covariance functions for litter size in …member: prof. dr. marija uremovic´ university of zagreb,...

UNIVERSITY OF LJUBLJANA

BIOTECHNICAL FACULTY

ZOOTECHNICAL DEPARTMENT

Zoran LUKOVIC

COVARIANCE FUNCTIONS FOR LITTER SIZE IN PIGS USING A RANDOMREGRESSION MODEL

DOCTORAL DISSERTATION

KOVARIAN CNE FUNKCIJE ZA VELIKOST GNEZDA PRI PRAŠI CIH VMODELU Z NAKLJU CNO REGRESIJO

DOKTORSKA DISERTACIJA

Ljubljana, 2006

Lukovic Z. Covariance functions for litter size in pigs using a random regression model.Doctoral Dissertation. Ljubljana, Univ. of Ljubljana, Biotechnical Faculty, Zootechnical Department, 2006 II

The present dissertation thesis is conclusion of a doctoralstudy. The research work was performed at

the Zootechnical Department of Biotechnical Faculty (Slovenia). Litter records from three farm were

supplied by the Slovenian pig breeding organization.

Senate of Biotechnical Faculty nominated Prof. dr. Milena Kovac as supervisor for this doctoral

dissertation.

Commission for evaluation and justification:

Chairperson: Prof. dr. Jurij POHAR

University of Ljubljana, Biotechnical Faculty, Zootech. Dept.

Member: Prof. Milena KOVAC, PhD

University of Ljubljana, Biotechnical Faculty, Zootech. Dept.

Member: Prof. dr. Marija UREMOVIC

University of Zagreb, Faculty of Agriculture, Croatia

Date of justification: July 10th 2006

The thesis is result of my own research work.

Zoran Lukovic

Lukovic Z. Covariance functions for litter size in pigs using a random regression model.Doctoral Dissertation. Ljubljana, Univ. of Ljubljana, Biotechnical Faculty, Zootechnical Department, 2006 III

KEY WORDS DOCUMENTATION

DN DdDC UDC 636.4.082.4:519.22:57.087(043.3)=20CX livestock production/genetic evaluation/random regression models/longitudinal

data/litter size/pigsCC AGRIS L10/U10/5300AU LUKOVI C, ZoranAA KOVA C, Milena (supervisor)PP SI-1230 Domžale, Groblje 3PB University of Ljubljana, Biotechnical Faculty, Zootechnical DepartmentPY 2005TI COVARIANCE FUNCTIONS FOR LITTER SIZE IN PIGS USING A RANDOM RE-

GRESSION MODELDT Doctoral DissertationNO XIV, 111 p., 30 tab., 19 fig., 117 ref.LA enAL en/slAB Litter records from three large farms were used. Each farmhad two data sets. The first one in-

cluded records from the first to the sixth parity, and the second data set included records fromthe first to the tenthparity. Sows had between 3.08 and 3.50 litters in the first data set, and in thesecond data set between 3.56 and 4.16 litters.Estimates of variances in repeatability model, co-variance components for multiple-trait analysis and for random-regression coefficients wereestimated by the REML method and the VCE5 software package. Fixed effects in modelfor the number of piglets born alive included sow genotype, parity, weaning to conception,mating season and service sire as class effects. In fixed partof the model previous lactationlength as linear regression and age at farrowing as quadratic regression were included. Modelfor NBA included direct additive genetic, permanent environmental and common litter envi-ronmental effect, except in multiple-trait model that did not include permanent environmentaleffect. Orthogonal Legendre polynomials (LG) of differentorder were fit to random effectsin random regression model. LG with four coefficients were found sufficient. Heritabilityestimates were around 10 % in repeatability model, and between 10 and 14 % in multiple-trait analysis and random regression model. Estimates of permanent environmental effectwere between 5 and 6 % in repeatability model, and between 0.04 and 0.10 in the randomregression model. In repeatability model the ratio for the common litter environmental vari-ance with respect to the total variance were between 1 and 2 %,in multiple-trait analysisand random regression model between 1 and 5 %. The eigenfunctions and correspondingeigenvalues showed that including higher parities in data sets two increased percentages ofthe total variability which were explained with individualproduction curve. In the seconddata sets around 85 to 90 % of the total genetic variability was explained by the constantterm in regression, while 10 to 15 % was genetic variability in the shape of litter size curve.

Lukovic Z. Covariance functions for litter size in pigs using a random regression model.Doctoral Dissertation. Ljubljana, Univ. of Ljubljana, Biotechnical Faculty, Zootechnical Department, 2006 IV

KLJUCNA DOKUMENTACIJSKA INFORMACIJA

ŠD DdDK UDK 636.4.082.4:519.22:57.087(043.3)=20KG živinoreja/genetski parametri/modeli z nakljucno regresijo/zaporedne

meritve/velikost gnezda/prašiciKK AGRIS L10/U10AV LUKOVI C, Zoran, mag., dipl. ing. agr.SA KOVAC, Milena (mentor)KZ SI-1230 Domžale, Groblje 3ZA Univerza v Ljubljani, Biotehniška fakulteta, Oddelek zazootehnikoLI 2005IN KOVARIAN CNE FUNKCIJE ZA VELIKOST GNEZDA PRI PRAŠICIH V MODELU Z

NAKLJUCNO REGRESIJOTD Doktorska disertacijaOP XIV, 111 str., 30 pregl., 19 sl., 117 vir.IJ enJI en/slAI Podatke o velikosti gnezda s treh farm smo analizirali tako, da smo jih razdelili glede na

število prasitev. V prvi niz podatkov smo vkljucili meritve gnezda od prve do šeste prasitve.Drugi niz je obsegal podatke od prve do desete prasitve. Število gnezd na svinjo se je razliko-valo in sicer so v prvem nizu podatkov imele svinje med 2.88 in3.50 gnezd. V drugem nizupodatkov je bila ta vrednost vecja in je variirala od 3.56 do 4.16 gnezd. Za oceno varianc vponovljivostnem modelu ter komponent kovarianc v veclastnostnem modelu in v modelu znakljucno regresijo smo uporabili metodo REML v statisticnem paketu VCE5. V model zaštevilo živorojenih pujskov smo vkljucili naslednje sistematske vplive z nivoji: genotip sv-inje, zaporedno prasitev, poodstavitveni premor, sezono pripusta in merjasca. V sistematskidel modela smo vkljucili še dolžino predhodne laktacije kot linearno regresijoin starost obprasitvi kot kvadratno regresijo. Nakljucni del modela za število živorojenih pujskov je vse-boval direktni aditivni genetski vpliv, vpliv permanentnega okolja in vpliv skupnega okolja vgnezdu. Veclastnostni model je bil brez vpliva permanentnega okolja.Ortogonalne Legen-drove polinome razlicnih stopenj smo uporabili za modeliranje nakljucnih vplivov v modeluz nakljucno regresijo in ugotovili, da je najprimernejši Legendrovpolinom tretje stopnje.Ocenjena heritabiliteta za velikost gnezda je bila 10 % v ponovljivostnem modelu, med 10 in14 % pa v veclastnostnem modelu in v modelu z nakljucno regresijo. Delež permanentnegaokoljskega vpliva je znašal med 5 in 6 % v ponovljivostnem modelu in med 4 in 10 % vmodelu z nakljucno regresijo. Delež skupnega okolja v gnezdu je zajel od 1 do2 % vari-abilnosti v ponovljivostnem modelu ter med 1 in 5 % v veclastnostnem modelu in modelu znakljucno regresijo. Lastne funkcije s pripadajocimi lastnimi vrednostmi so pokazale, da sez vkljucitvijo višjih zaporednih prasitev v podatke povecuje delež skupne variabilnosti po-jasnjene z individualno proizvodno krivuljo. V podatkih, kjer smo vkljucevali tudi meritveod sedme do desete prasitve, je med 85 in 90 % skupne genetske variabilnosti pojasnil kon-stantni clen, med tem ko oblika proizvodne krivulje za velikost gnezda pojasni med 10 in15 % genetske variabilnosti.

Lukovic Z. Covariance functions for litter size in pigs using a random regression model.Doctoral Dissertation. Ljubljana, Univ. of Ljubljana, Biotechnical Faculty, Zootechnical Department, 2006 V

TABLE OF CONTENTS

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Key words documentation (KWD) III

Klju cna dokumentacijska informacija (KDI) IV

Table of contents V

List of Tables VIII

List of Figures XII

Abbreviations and symbols XV

1 INTRODUCTION 1

2 LITERATURE REVIEW 3

2.1 SELECTION FOR LITTER SIZE 3

2.2 FACTORS AFFECTING LITTER SIZE IN PIGS 5

2.2.1 Age at farrowing 5

2.2.2 Lactation length and weaning to conception interval 6

2.2.3 Season 7

2.2.4 Genotype 8

2.2.5 Direct additive genetic effect 11

2.2.6 Maternal additive genetic effect 13

2.2.7 Sire effect 14

2.2.8 Common litter environmental effect 15

2.2.9 Permanent environmental effect 16

2.3 GENETIC CORRELATIONS 16

2.4 STATISTICAL MODELS FOR GENETIC EVALUATION OF LITTER

SIZE 18

2.4.1 Repeatability model 22

2.4.2 Multiple-trait model 22

2.4.3 Random regression model 23

3 MATERIAL AND METHODS 25

3.1 MATERIAL 25

3.2 DATA FILE ORGANIZATION 30

Lukovic Z. Covariance functions for litter size in pigs using a random regression model.Doctoral Dissertation. Ljubljana, Univ. of Ljubljana, Biotechnical Faculty, Zootechnical Department, 2006 VI

3.3 METHODS 31

3.3.1 Model development 31

3.3.2 Implemented models 33

3.3.3 Models in matrix notation and covariance structure 35

3.3.4 Eigenvalues and eigenfunctions 39

3.3.5 Breeding values 39

4 RESULTS 40

4.1 MODEL SELECTION 40

4.1.1 Fixed effects 40

4.1.2 Age at farrowing and parity 42

4.1.3 Service sire 46

4.1.4 Weaning to conception interval 47

4.1.5 Mating season 50

4.1.6 Previous lactation length 53

4.1.7 Sow genotype 54

4.1.8 Random effects 55

4.2 COMPUTATIONAL REQUIREMENTS 56

4.3 REPEATABILITY MODEL 56

4.4 MULTIPLE-TRAIT ANALYSIS 58

4.4.1 Variance components 58

4.4.2 Correlations 61

4.5 RANDOM REGRESSION MODEL 63

4.5.1 Eigenvalues and eigenfunctions 63

4.5.2 Covariance components 66



5 DISCUSSION 81

5.1 CHOICE OF THE MODEL AND ESTIMATION OF FIXED EFFECTS 81

5.2 COMPUTATION REQUIREMENTS 84

5.3 REPEATABILITY MODEL 85

Lukovic Z. Covariance functions for litter size in pigs using a random regression model.Doctoral Dissertation. Ljubljana, Univ. of Ljubljana, Biotechnical Faculty, Zootechnical Department, 2006 VII

5.4 MULTIPLE-TRAIT ANALYSIS 86



5.5 RANDOM REGRESSION MODEL 86

5.5.1 Eigenvalue analysis 87




6 CONCLUSIONS 89

7 SUMMARY (POVZETEK) 90

7.1 SUMMARY 90

7.2 POVZETEK 95

8 REFERENCES 103

ACKNOWLEDGEMENTS

ZAHVALA

Lukovic Z. Covariance functions for litter size in pigs using a random regression model.Doctoral Dissertation. Ljubljana, Univ. of Ljubljana, Biotechnical Faculty, Zootechnical Department, 2006 VIII

LIST OF TABLES

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Table 1: Number of piglets born alive by different genotypes

Tabela 1: Število živorojenih pujskov pri razlicnih genotipih 10

Table 2: Heritability estimates for number of piglets born alive from the literature

Tabela 2: Ocene heritabilitete za število živorojenih pujskov iz literature 12

Table 3: Genetic correlation for the number of piglets born alive between succes-

sive parities

Tabela 3: Primerjava genetskih korelacij za število živorojenih pujskov med za-

porednimi prasitvami 17

Table 4: Fixed effects in the models for the number of pigletsborn alive

Tabela 4: Pregled sistematskih vplivov v modelih za številoživorojenih pujskov 20

Table 5: Elimination criteria for age at farrowing by parity

Tabela 5: Kriterij za izlocitev podatkov zaradi starosti ob prasitvi po zaporednih

prasitvah 25

Table 6: Data structure for data sets DS1 and DS2 for farms A, B, and C

Tabela 6: Struktura podatkov za niza podatkov DS1 in DS2 po farmah 26

Table 7: Number of animals, mean (x) and standard deviation (σ) for number of

piglets born alive (NBA), age at farrowing (AF), previous lactation length

(PLL) and mean and mode for weaning to conception interval (WCI) in

data sets DS1 and DS2 within farms

Tabela 7: Število živali, povprecje in standardni odklon za število živorojenih pu-

jskov, starost ob prasitvi, dolžino predhodne laktacije ter povprecje in

modus za poodstavitveni premor v nizih podatkov DS1 in DS2 pofarmah 27

Table 8: Number of records, mean and standard deviation for the number of piglets

born alive (NBA), previous lactation length (PLL) and mean and mode for

weaning to conception interval (WCI) by sow genotype and farm for data

set DS2

Tabela 8: Število meritev, povprecje in standardni odklon za število živorojenih pu-

jskov (NBA), dolžino predhodne laktacije (PLL) ter povprecje in modus

za poodstavitveni premor (WCI) po genotipu svinje po farmahv nizu po-

datkov DS2 28

Table 9: Pedigree structure within farms

Lukovic Z. Covariance functions for litter size in pigs using a random regression model.Doctoral Dissertation. Ljubljana, Univ. of Ljubljana, Biotechnical Faculty, Zootechnical Department, 2006 IX

Tabela 9: Struktura porekla po farmah 30

Table 10: Representation of prepared data structure

Tabela 10: Izsek iz pripravljenih podatkov 30

Table 11: Modelling of the fixed effects

Tabela 11: Modeliranje sistematskih vplivov 32

Table 12: Coefficients of determination (R2) and degrees of freedom (d.f.) for dif-

ferent models per farm

Tabela 12: Koeficienti determinacije (R2) in stopnje prostosti (d.f.) za razlicne mod-

ele po farmah 41

Table 13: Number of levels for mating season and service sireeffect, degrees of

freedom (d.f.) for model, coefficient of determination (R2) and standard

deviation (σ) on three farms

Tabela 13: Število nivojev za sezono pripusta in merjasca oceta gnezda, stopnje pros-

tosti za model (d.f.), koeficient determinacije (R2) in standardni odklon

(σ) po farmah 41

Table 14: Change in the coefficient of determination (∆R2) from theR2 obtained

with full model for individual effect per farm

Tabela 14: Sprememba koeficienta determinacije (∆R2) od koeficienta determi-

nacije polnega modela za posamezne vplive po farmah 42

Table 15: Estimates of regression coefficients with standard errors (SEE) for age at

farrowing nested within parity class per farm

Tabela 15: Ocene regresijskih koeficientov in njihove standardne napake (SEE) za

starost ob prasitvi po zaporednih prasitvah in farmah 45

Table 16: Effect of weaning to conception interval (WCI) on litter size expressed as

a deviation from day 5 per farm

Tabela 16: Vpliv poodstavitvenega premora (WCI) na velikost gnezda kot

odstopanje od petega dneva po farmah 49

Table 17: Estimates of linear regression coefficient with standard errors (SEE) for

lactation length per farm

Tabela 17: Ocene linearnih regresijskih koeficientov in njihove standardne napake

(SEE) za dolžino laktacije po farmah 53

Table 18: Comparison among sow genotypes on three farms

Tabela 18: Ocene razlik* med genotipi svinj s standardnimi napakami (SEE) po

farmah 54

Lukovic Z. Covariance functions for litter size in pigs using a random regression model.Doctoral Dissertation. Ljubljana, Univ. of Ljubljana, Biotechnical Faculty, Zootechnical Department, 2006 X

Table 19: Estimates of (co)variance matrices with standarderrors of estimate in the

model with maternal additive genetic effect on three farms

Tabela 19: Ocene matrik kovarianc sa standardnimi napakamiocen v modelu z ma-

ternalnim aditivnim genetskim vplivom 55

Table 20: Computational requirements in repeatability model (REP), multiple-trait

model (MTM) and random regression model (RRM) with different order

of Legendre polynomials (LG1–LG3) for DS1 and DS2 for farm B

Tabela 20: Poraba racunalniških kapacitet v ponovljivostnem modelu (REP),

veclastnostnem modelu (MTM) in modelu z nakljucno regresijo (RRM)

z razlicnim stopnjam Legendrovih polinomov (LG1–LG3) za nize po-

datkov DS1 in DS2 pri farmi B 56

Table 21: Estimates of variance components and ratios of phenotypic variance for

the number of piglets born alive by farms with repeatabilitymodel for

data set DS2

Tabela 21: Ocene komponent varianc in deleži fenotipske variance za število živo-

rojenih pujskov po farmah v ponovljivostnem modelu za niz podatkov

DS2 57

Table 22: Estimated variance components with standard errors of estimates in

multiple-trait model

Tabela 22: Ocene komponent variance in standardne napake ocen v veclastnostnem

modelu 60

Table 23: Estimates of direct additive genetic (above diagonal) and phenotypic cor-

relations (below diagonal) by multiple-trait model for data set DS1

Tabela 23: Ocene direktnih aditivnih genetskih (nad diagonalo) in fenotipskih ko-

relacij (pod diagonalo) v veclastnostnem modelu za niz podatkov DS1 61

Table 24: Estimates of common litter environmental (above diagonal) and residual

(below diagonal) correlations by multiple-trait model fordata set DS1

Tabela 24: Ocene korelacij za skupno okolje v gnezdu (nad diagonalo) in za ostanek

(pod diagonalo) v veclastnostnem modelu za niz podatkov DS1 62

Table 25: Eigenvalues of estimated covariance matrices of random-regression coef-

ficients with proportion (in parenthesis) of the total variability for random

effects in data sets DS1 with different order of Legendre polynomials

(LG1 – LG3)

Tabela 25: Lastne vrednosti za ocenjene matrike kovarianc za nakljucne regresijske

koeficiente z deležem (v oklepajih) v celotni varianci za nakljucne vplive

v nizih podatkov DS1 za razlicne stopnje Legendrovih polinomov (LG1

– LG3) 64

Lukovic Z. Covariance functions for litter size in pigs using a random regression model.Doctoral Dissertation. Ljubljana, Univ. of Ljubljana, Biotechnical Faculty, Zootechnical Department, 2006 XI

Table 26: Eigenvalues of estimated covariance matrices of random regression coef-

ficients with proportion (in parenthesis) of the total variability for random

effects in data sets DS2 with different order of Legendre polynomials

(LG1–LG3)

Tabela 26: Lastne vrednosti za ocenjene matrike kovarianc za nakljucne regresijske

koeficiente z deležem (v oklepajih) v celotni varianci za nakljucne vplive

v nizu podatkih DS2 za razlicne stopnje Legendrovih polinomov (LG1–

LG3) 65

Table 27: Estimated variance components and proportions inthe phenotypic varia-

tion for NBA in random regression model with cubic Legendre polyno-

mial for three farms in data set DS1

Tabela 27: Ocene komponent variance in deleži v fenotipski varianci za model z

nakljucno regresijo z uporabljenim Legendrovim polinomom tretjestop-

nje po farmah za niz podatkov DS1 67

Table 28: Estimated variance components and proportions ofphenotypic variation

for NBA in random regression model with cubic Legendre polynomial

for the three farms in data set DS2

Tabela 28: Ocene komponent variance in deleži od fenotipskevariance v modelu z

nakljucno regresijo z uporabljenim Legendrovim polinomom tretjestop-

nje po farmah za niz podatkov DS2 71

Table 29: Estimates of direct additive genetic (above diagonal) and phenotypic cor-

relations (below diagonal) by RRM for data set DS2

Tabela 29: Ocene direktnih aditivnih genetskih (nad diagonalo) in fenotipskih ko-

relacij (pod diagonalo) v modelu z nakljucno regresijo v nizu podatkih

DS2 75

Table 30: Estimates of residual common litter environment (above diagonal) and

permanent environment correlations (below diagonal) by RRM for data

set DS2

Tabela 30: Ocene korelacij za skupno okolje v gnezdu (nad diagonalo) in za per-

manentno okolje (pod diagonalo) v modelu z nakljucno regresijo v nizih

podatkov DS2 77

Lukovic Z. Covariance functions for litter size in pigs using a random regression model.Doctoral Dissertation. Ljubljana, Univ. of Ljubljana, Biotechnical Faculty, Zootechnical Department, 2006 XII

LIST OF FIGURES

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Figure 1: Relationship between genotypes

Slika 1: Razmerje med genotipi 9

Figure 2: Structure of random effects

Slika 2: Struktura nakljucnih vplivov 18

Figure 3: Relative frequency of sows per common litter by farm

Slika 3: Relativna frekvenca svinj na gnezdo po farmah 27

Figure 4: Averages and phenotypic variances for the number of piglets born alive

by parity

Slika 4: Povprecja in fenotipske variance za število živorojenih pujskov po za-

porednih prasitvah 29

Figure 5: Relationship between age at farrowing and the number of piglets born

alive nested within parity for farm B

Slika 5: Povezava med starostjo ob prasitvi in številom živorojenih pujskov zno-

traj zaporedne prasitve na farmi B 43

Figure 6: Relationship between the age at farrowing and the number of piglets born

alive nested within three parity class per farm

Slika 6: Povezava med starostjo ob prasitvi inštevilom živorojenih pujskov znotraj

razredov zaporedne prasitve po farmah 44

Figure 7: Estimates of the service sire effect on the number of piglets born alive for

farm B

Slika 7: Ocene za vpliv merjasca - oceta gnezda za število živorojenih pujskov na

farmi B 46

Figure 8: The average number of piglets born alive and distribution of weaning to

conception interval per farm

Slika 8: Povprecno število živorojenih pujskov in porazdelitev poodstavitvenega

premora po farmah 48

Figure 9: Parametrical function for description of relationship between weaning to

conception interval and the number of piglets born alive

Slika 9: Parametricna funkcija za opis povezave med poodstavitvenim premorom

in številom živorojenih pujskov 50

Lukovic Z. Covariance functions for litter size in pigs using a random regression model.Doctoral Dissertation. Ljubljana, Univ. of Ljubljana, Biotechnical Faculty, Zootechnical Department, 2006 XIII

Figure 10: Estimates of mating season effect on number of piglets born alive per

farm

Slika 10: Ocene vpliva sezone pripusta na število živorojenih pujskov po farmah 52

Figure 11: Relationship between the number of piglets born alive and previous lac-

tation length per farm

Slika 11: Povezava med dolžino predhodne laktacije in številom živorojenih pu-

jskov po farmah 53

Figure 12: Estimated genetic eigenfunctions from random regression model with cu-

bic power for number of piglets born alive on farm B

Slika 12: Ocenjene genetske lastne funkcije za Legendrov polinom tretje stopnje v

modelu z nakljucno regresijo za število živorojenih pujskov na farmi B 66

Figure 13: Comparison of ratios in the phenotypic variance with random regression

model (lines) and multiple-trait model (triangles) for data set DS1 over

parities per farm

Slika 13: Primerjave deležev od fenotipske variance v modelu z nakljucno regresijo

(crte) in v veclastnostnem modelu (trikotniki) za niz podatkov DS1 po

zaporednih prasitvah po farmah 69

Figure 14: Phenotypic variances and proportions of the phenotypic variance over

parities with Legendre polynomials of the cubic power for the number of

piglets born alive

Slika 14: Fenotipske variance in deleži od fenotipskih varianc po zaporednih pra-

sitvah z uporabljenim Legendrovim polinomom tretje stopnje za število

živorojenih pujskov 72

Figure 15: Comparison of estimates for ratios in the phenotypic variance between

data sets DS1 and DS2 per farm

Slika 15: Primerjava ocen deležev fenotipske variance med nizi podatkov DS1 in

DS2 po farmah 74

Figure 16: Genetic correlation for the number of piglets born alive between parities

using RRM for farm B

Slika 16: Genetske korelacije za število živorojenih pujskov med zaporednimi pra-

sitvami v modelu z nakljucno regresijo za farmo B 76

Figure 17: Estimated breeding values for number of piglets born alive over parities

for seven sires

Slika 17: Napovedi plemenskih vrednosti za število živorojenih pujskov po za-

porednih prasitvah pri sedmih merjascih 78

Lukovic Z. Covariance functions for litter size in pigs using a random regression model.Doctoral Dissertation. Ljubljana, Univ. of Ljubljana, Biotechnical Faculty, Zootechnical Department, 2006 XIV

Figure 18: Phenotypic averages for the number of piglets born alive over parities for

seven sires

Slika 18: Fenotipska povprecja za število živorojenih pujskov po zaporednih pra-

sitvah pri sedmih merjascih 79

Figure 19: Deviations from the phenotypic average for the number of piglets born

alive over parities for seven sires

Slika 19: Odstopanja od fenotipskega povprecja za število živorojenih pujskov po

zaporednih prasitvah pri sedmih merjascih 80

Lukovic Z. Covariance functions for litter size in pigs using a random regression model.Doctoral Dissertation. Ljubljana, Univ. of Ljubljana, Biotechnical Faculty, Zootechnical Department, 2006 XV

ABBREVIATIONS AND SYMBOLS

AF Age at farrowing

BLUP Best linear unbiased prediction

d.f. Degrees of freedom

DS Data set

GLM General linear model

LGn Legendre polynomial of order n

MTM Multiple-trait model

MMM Mixed model methodology

NBA Number of piglets born alive

PLL Previous lactation length

REML Restricted maximum likelihood

RRM Random regression model

R2 Coefficient of determination

WCI Weaning to conception interval

⊗ Kronecker product∑

L

Direct sum

Lukovic Z. Covariance functions for litter size in pigs using a random regression model.Doctoral Dissertation. Ljubljana, Univ. of Ljubljana, Biotechnical Faculty, Zootechnical Department, 2006 1

1 INTRODUCTION

Selection in pigs is conducted for many years to increase growth, carcass merit and sow productivity.

After an efficient improvement of growth and carcass traits,commercial breeding programmes placed

a great emphasis on improving reproductive traits for maternal breeds and lines. Many analyses

confirmed that genetic advance in overall reproductive efficiency can be attained most effectively by

selection for litter size. Litter size is relatively easy tomeasure, thus including this trait in a selection

programmes is often warranted.

Improvement of litter size in pigs through selection was considered a difficult task in the past. Re-

productive traits were not included prominently in commercial breeding programmes in the past due

to slow improvement by traditional selection techniques incontrast to growth and carcass traits. Low

heritability, negative correlation between direct additive genetic and maternal effect, limited size of

the nucleus population, expression only in females, and relatively late age of measurements, were the

main reasons for slow improvement. From these reasons, selection for litter size was considered a

long time as noneffective. On the other hand, existence of large direct additive genetic variance for

litter size and availability of records on many relatives, provides possibility for successful selection

on litter size. Dividing pig breeding programmes into specialized sire and dam lines resulted in a

higher emphasis on selection for litter size in the dam lines. Worldwide, pig breeders actually proved

that selection for litter size can be successful. An important step was the development in computer

technology and use of the mixed model methodology (MMM). Theuse of the MMM has become a

standard in most farm animal species to estimate dispersionparameters by residual maximum like-

lihood (REML) method and to predict breeding values of animals by best linear unbiased prediction

(BLUP). It provides simultaneously estimates of genetic and environmental parameters, taking into

account the relationship among animals. Although, selection for traits with low heritability often re-

lies on use of molecular genetic, selection based on MMM and numerous measurements is still very

powerful and cost effective.

Litter size is a complex trait prenatally combined from the following components: ovulation rate,

embryo survival, and uterine capacity. Postnatal litter size can be described as number of piglets born

total, number of piglets born alive, and number of weaned piglets. Number of piglets at weaning is

even of greater commercial importance than litter size at birth, but selection for it is impossible under

conditions of crossfostering. High genetic correlation between number of piglets born and number of

piglets born alive makes selection on only one of them sufficient. Selection for number of piglets born

also increases number of stillborn piglets. Therefore, number of piglets born alive is the selection trait

of choice in improvement of litter size in most breeding programmes.

Number of piglets born alive is affected by numerous environmental and genetic factors and inter-

actions between them. On large farms some effects were recorded. Data recording used firstly for

management monitoring and for pedigree control. These datawere also used for selection purposes.

Data on sow fertility provide satisfactory description of effects for litter size and prediction of breed-

ing values using MMM. Fixed part of the model for genetic evaluation of litter size includes often


effects as parity, different description of mating or farrowing season, service sire and breed of service

sire, age at farrowing, and different reproductive intervals which affect litter size as well as over-

all efficiency of sow production. Most frequently used reproductive intervals are lactation length,

weaning to conception interval, and farrowing interval. Beside direct additive genetic effect used for

breeding value estimation, random part of the models for litter size also includes other genetic and

environmental effects provided by data. Scientists in the area of animal genetics are trying to develop

statistical models for litter size, that also include othergenetic effects (additive maternal, paternal and

non-additive genetic effect as dominance). In the past, these effects were not included in models,

mainly because of small computer power and lack of general software packages. The most common

used environmental effects are permanent and common littereffect. Genetic evaluation in litter size

is in many countries based on repeatability model. Low genetic correlations between litter size in

different parities is reason for use of multiple-trait analysis, although it is rarely used. In the recent

period, study of random regression models in different areas of animal production increased. Random

regression models are especially suitable for longitudinal traits that change over time. Litter size is

measured more than once in a sow lifetime and can be consider as longitudinal trait too. There are

ideas for using random regression model in modeling of discrete traits, where litter size belongs.

The aims of thesis were:

- to develop appropriate fixed part of the model for genetic evaluation for litter size,

- to determine covariance functions for number of piglets born alive,

- to estimate genetic and environmental parameters using a repeatability model, multiple-trait analysis

and random regression and compare them,

- to check possibility for using random regression in genetic evaluation of litter size.


2 LITERATURE REVIEW

2.1 SELECTION FOR LITTER SIZE

Early experiments on selection for litter size were successful only marginally. Many of these exper-

iments produced no significant genetic or phenotypic trends. Several selection methods were used.

According to literature the method of independent culling level was mainly used. A certain level of

litter size was established, and all individuals below thatlevel were culled. Selection experiments

based on an independent culling level in Sweden (Johansson,1981) did not improve litter size in the

period of twenty years. Short duration of experiments, small population sizes as well as existence

of maternal effects, which may be negatively correlated to direct genetic effect, were reported as the

reasons for non significant trend from direct litter size selection (Vangen, 1981). Little or no response

have been also explained by a low heritability for litter size and a failure to achieve sufficient selection

intensity (Ollivier, 1982), as well as by low genetic correlations between parities (Bolet et al., 1989).

Results like these discouraged pig producers from including litter size in selection objectives. Thus,

there was insufficient selection pressure on litter size in breeding herds until the late 1980s.

Selection index method implemented by Hazel (1943) presents the overall net merit of the individual

considering several traits of economic importance. It provided a superior selection criterion compared

to other forms of selection including single trait selection and multiple-trait selection via independent

culling level used before. Litter size has been included to selection index since the 60’s. In the middle

of the 1970’s in USA, litter size started to improve using SowProductivity Index which included

the number of piglets born alive and 21 day litter weight. In gilts, Neal and Irvin (1992) achieved a

difference of 1.43 liveborn piglets between the select and control line after ten generations of selection

on Sow Productivity Index. Family selection was introducedin the dam lines to select for litter size

(Avalos and Smith, 1987; Haley et al., 1988) by the end of the 80’s. Family selection was a selection

method where superior families rather than superior individuals were chosen for breeding. High

rate of genetic improvement was expected theoretically selecting on family index (Avalos and Smith,

1987). In a deterministic study, they showed that given a heritability of 0.10 and the use of all family

information in selection, an improvement of litter size of 0.50 piglets per year could be expected.

Indirect selection on litter size was studied for the ovulation rate, embryonic survival, and uterine ca-

pacity. Selection for ovulation rate resulted in a correlated increase in litter size, and the obtained

difference was maintained during a period of relaxed selection (Cunningham et al., 1979). Nine gen-

erations of selection for higher ovulation rate were followed by two generations of random selection

and then eight generations of selection for increased litter size at birth, decreased age at puberty, or

continued random selection in the high ovulation rate line (Lamberson et al., 1991). Estimate of re-

sponse to selection for litter size was 1.06 piglets per litter (P<0.01) with regression on generation

number, and 0.48 piglets per litter with animal model. Cumulative response in litter size to selec-

tion for the ovulation rate and then litter size was 1.8 and 1.4 piglets per litter estimated by the two

methods. Johnson and Cassady (1998) attempted to improve litter size with ten generations of index


selection for increased ovulation rate and embryonal survival followed by one generation of random

selection and three generations of litter size selection. Response at end of the experiment was 1.2

piglets born alive (P<0.05). Gama and Johnson (1993) reported selection response of one piglet after

eight generations of selection for litter size, following selection on ovulation rate. The conclusion was

that selection for ovulation rate would improve litter size, but selection was very inefficient as only

20 to 30 % of the increase in ovulation rate was expressed in litter size (Gama and Johnson, 1993).

Furthermore, these selection experiments require expensive investments of time and equipment, and

are not practical for most selection programmes (Johnson, 1992).

Hyperprolific selection scheme was an effective way of improving litter size. Legault and Gruand

(1976) proposed to use large-scale computerized recordingsystems in order to identify exceptionally

prolific sows, so called ’hyperprolific’ sows. By repeated backcrosses to hyperprolific sows, the ge-

netic merit of their boar progenies was progressively raised to the level of the dams. The selection

scheme has been successfully applied to the French Large White population (Bidanel et al., 1994).

In a hyperprolific Large White experimental herd, the average litter size of hyperprolific sows ex-

ceeded the normal Large White contemporaries by 2.6 piglet per litter born and 1.5 piglet per litter

born alive. Although the experiment resulted in an initial increase on the female side of one piglet,

continuous screening of hyperprolific females was less effective because multiplication phase took

many generations to obtain a sizable population of descendants to start a new round of intense selec-

tion (Avalos and Smith, 1987). Therefore, hyperprolific selection scheme can be useful for setting

up a nucleus herd from a previously unselected population, but not for achieving high continuous

response. Overall annual genetic gain was much smaller, because of backcrossing. Noguera et al.

(1998) showed that selection for litter size could be successful, if a hyperprolific breeding scheme

and mixed model methodology were combined. The average estimated difference between selected

and control line was 0.57 piglets born alive (P<0.05) in favour of line selected for prolificacy. Bolet

et al. (2001) reported about selection experiment on littersize for seventeen generation. Selection was

performed for eleven generations in a closed line, and afterthat period, the selected line was opened

to gilt daughters of hyperprolific boars and sows. After eleven generations of selection in a closed

line, response in litter size was not significant. The total genetic gain of about 1.4 piglets per litter

at birth was a consequence of migration (0.8 piglets per litter) and a within-line selection (0.6 piglets

per litter).

Mixed model methodology in litter size has been used since the late 80’s due to improved knowledge

of quantitative genetics and computer development. Application of mixed model methodology using

residual maximum likelihood method and best linear unbiased predictor was an important step, re-

sulted in clear genetic progress for litter size in dam lines. In their theoretical study Belonsky and

Kennedy (1988) have pointed out that higher genetic trends could be obtained by using mixed model

methodology with animal model than traditional selection index procedures, especially for traits with

low heritability as litter size. In a closed pig herd, response to selection on best linear unbiased pre-

dictor was greater than selection on phenotype selection by55 % (without additional culling) to 81 %

(with additional culling). Similar results were also obtained by Sorensen (1988). Selection response


for a single trait with heritability of 10 % was up to 25 % smaller in selection index than in animal

model. Smaller response using selection index in relation to response obtained with animal model

Sorensen (1988) assigned to the sources of bias introduced in the construction of the selection index

owing to genetic trend and the smaller accuracy of the selection index relative to animal model. Lof-

gren et al. (1994) obtained an increase of approximately 0.1liveborn piglets per litter per year in the

period from 1987 to 1992. Genetic response after nine generations of selection estimated breeding

value for the number of piglets born alive using the animal model differed by 0.08 piglets/year be-

tween the control and selected line (Holl and Robison, 2002). The result was 0.12 piglets per year in

phenotypic trend.

2.2 FACTORS AFFECTING LITTER SIZE IN PIGS

Many environmental and genetic factors, as well as complex interactions between them, influence

litter size in pigs. They could be arranged into two major groups (Clark and Leman, 1986a). The first

group includes factors which are often recorded by commercial pig producers and contains effects

like parity, sow breed or genotype, age at farrowing, lactation length, weaning to conception interval

etc. The data were used for management and selection purposes. The second group includes factors

such as husbandry practices, nutrition, and diseases. Although they are very important, data on these

factors are not always available and often cannot be evaluated.

2.2.1 Age at farrowing

Age at farrowing can be described in two different ways: chronologically and physiologically.

Chronological age is expressed by age in days, months, or years, while physiological age is used

to indicate maturation process. In gilts, it is expressed bythe number of oestruses before the first

mating, while in sows by parities.

In gilts, litter size increases with age at the first mating. The relationship is explained as a conse-

quence of higher ovulation rate in later oestruses (Brooks and Smith, 1980). More recently, results

by Tummaruk et al. (2001) showed that a 10 day increase in age at first mating in gilts resulted in an

increase (P<0.001) of about 0.1 piglet born alive. In the same study, they also reported the effect of

age at first mating on litter size in higher parities. A 10 day increase in age at first mating resulted

in a decrease (P<0.05) in litter size in sows in parities 4 and5. Usually, the second or third oestrus

is prefered for the first mating of gilts what coincides with age at first mating between 210 and 240

days. Clark et al. (1988) reported that age at conception didnot influence litter size after 245 days of

age. In other words, litter size in gilts was increasing up toone year. Later, litter size had a tendency

to be on the same level or was slowly decreasing. Southwood and Kennedy (1991) showed a signif-

icant increase in litter size for 0.18 piglets per month. Although age at farrowing had a significant

influence on first parity litter size, Culbertson and Mabry (1995) stated that benefit in increased litter

size greatly decreased in the second parity and was non-existent in any later parities.


In sows, litter size changes with parity. After the first parity, it increases gradually to a maximum in the

third to fifth parity and slowly decreases through higher parities (Koketsu and Dial, 1997; Tummaruk

et al., 2000a). Lower ovulation rate and smaller uterine capacity in gilts (Gama and Johnson, 1993)

as well as the fact that gilts and younger sows are more sensitive to environmental factors (Clark

et al., 1988) than older sows are the possible reasons for smaller litter size in the first few parities.

Changes in ovulation rate and uterine capacity with increasing parity (Gama and Johnson, 1993)

and sow ageing may have contributed to the parity influence onlitter size. Decrease in litter size

was especially noticed after the seventh parity (Koketsu and Dial, 1997). Investigation of late parity

decline in litter size would be of great importance in order to determine culling policy and give a

possibility for selection on the persistency of litter size.

The combination of age at farrowing and parity is the way to present interaction between these two

effects. Age at farrowing and parity cointeract, especially in the first litters. Differences in the age

when gilts reach puberty, as well as pregnancy failure in gilts and sows result in considerable variation

in the age of sows within parity. Because of the wide range of possible ages within parity and the fact

that litter size depends on age, there is a suggestion to combine those two effects. Interaction of age

at farrowing and parity was presented in the study of Triboutet al. (1998) and Marois et al. (2000).

2.2.2 Lactation length and weaning to conception interval

Lactation length is one of the effects determined by management. In large scale farms lactation

lasts mainly between three and four weeks. This is a sufficient period for uterine involution to be

completed. Lactation length of less than 21 days is related to incomplete involution of the uterine

endometrium and higher embryo mortality (Varley and Cole, 1976). Lactation length influence litter

size at the subsequent parity. Shorter lactation is associated with a smaller litter size at the subsequent

farrowing (Xue et al., 1993; Dewey et al., 1994). Tantasuparuk et al. (2000) reported no significant

effect of lactation length on subsequent litter size in tropical conditions. As possible reasons they

referred to a low variation in lactation length and lower reproductive efficiency under tropical climate

conditions. Babot et al. (1994) reported increase of litterby 0.03 to 0.04 piglets born alive for each

day of previous lactation length. Those results are in agreement with results by Xue et al. (1993).

Smaller regression coefficient was obtained in study from Logar et al. (1999). Higher estimates were

found in studies where records with short lactation length were excluded from analysis. Kovac et al.

(1983) got an increase of 0.057 piglets per litter per day when records with lactation length under 18

days were discarded. Marois et al. (2000) noticed that a linear regression on previous lactation length

could predict litter size well, except in case of lactation length shorter than 7 days.

Acceptance of recent directives of the European Communities (EC No 316/36 2001, 2001) that order

a minimum length of lactation of 28 days on large farms could decrease adverse effect of very short

lactation on litter size. On the other side, sows with lactation length longer than 28 days in inap-

propriate physiological state (exhaustion) due to suboptimal nutrition and increased requirements for

milk production could be the reason for smaller litter size in subsequent parity.


Weaning to conception interval has an obvious influence on the subsequent litter size. Fahmy et al.

(1979) reported an increase in litter size as the weaning to conception interval increased. They stated

that there is a progressive increase in litter size with the delay of oestrus after 7 days. Babot et al.

(1994) and Logar et al. (1999) assumed that the effect of previous weaning to conception interval on

litter size is linear and the found linear regression coefficient ranged between 0.0002 and 0.0074. On

the other hand, Dewey et al. (1994) found that relationship between previous weaning to conception

interval and litter size can be presented with U - shaped curve with litter size at a minimum for sows

conceiving at 7 to 10 days after weaning. More recent study byMarois et al. (2000) also showed that

the effect is curvilinear with lowest litter size for previous weaning to conception interval between

7th to 10th day and can not be accommodated by linear, quadratic regression or any other common

regressions.

The specific U - shaped curve obtained using segmented polynomials Marois et al. (2000) explained

by the following hypothesis. Sows with the shortest weaningto conception interval are those that

show oestrus most quickly after weaning because they are in agood nutritional and physiological

state (ten Napel et al., 1995). Because of a good state, they have larger litter size than sows conceiving

later. The lowest litter size for previous weaning to conception interval between 7 to 10 days could

be because these litters arise from late oestrus in sows in a poorer physiological state. Furthermore,

the decrease of litter size could be a consequence of inappropriate management (Kemp and Soede,

1996). Mating in a suboptimal mating time resulted in a smaller litter size (Nissen et al., 1997). At

still longer weaning to conception interval, larger littersizes could be explained by conception at the

second oestrus, which was generally recognized to give larger litter size than conception at the first

oestrus. Tummaruk et al. (2000b) showed the difference in litter size between sows with a fourth day

weaning to service interval and those mated on a tenth day after weaning. They also reported that

litter size increased as weaning to service interval increased from 10 to 20 days.

Weaning to conception interval decreases with the previouslactation length increase. Clark and

Leman (1984) found that early weaning of piglets had no effect on subsequent litter size when the

weaning to conception interval was greater than 14 days. However, when weaning to conception

interval in those sows was less than 14 days, litter size reduced by 0.1 piglets per each day between

the weaning age of 21 and 28 days. These findings showed the effects of lactation length on litter

size are likely to be influenced by weaning to conception interval and, in order to avoid bias, previous

lactation length and weaning to conception interval shouldbe considered together (Clark and Leman,

1986b).

2.2.3 Season

Season effect on litter size is mainly presented as the effect of mating or rarely as farrowing season.

The effect of season on litter size usually explains two sources of variation. Firstly, there are long-

term changes as a consequence of better environment, management, and selection. Secondly, litter

size oscillates due to short-term changes usually related to climate, as well as changes in technology


practices and other unknown sources of variation. The effect of season connected with climate can

be divided into the effects of photoperiod and temperature as reviewed by Clark and Leman (1986a).

The effects of photoperiod on litter size have not been adequately studied and often confounded with

other effects. Better known is temperature effect.

The effect of mating season on litter size is controversial.In many parts of the world, seasonal

variation in pig herds is characterized in some cases by a decrease in litter size during some part

of the year. Increased ambient temperature at mating time oncommercial farms in warm temperate

climatic zone in Australia showed to cause embryonic mortality (Love, 1979). The study by Paterson

et al. (1978) when mean daily maximum temperature exceeded thermoneutral temperature suggested

that heat stress imposed during early pregnancy caused lossof all embryos and the sow return to

oestrus. Therefore, litter size of sows conceived during summer was not significantly different from

sows conceiving at other seasons. In a temperate climate in Sweden, Tummaruk et al. (2000a) also

reported no seasonal variation of litter size during three year period. On the other hand, Koketsu and

Dial (1997) detected small litters in sows mated in hot summer months in southern Minnesota (USA).

Effect of season or more specifically, ambient temperature on litter size may be partly confounded

by other effects, for example by effect of prolonged weaningto conception interval. Britt et al.

(1983) stated that sows which weaned in summer took longer toreturn to oestrus than sows which

weaned at other seasons. In addition, sows during summer hadshorter interval from onset of oestrus

to ovulation and higher possibility to fail optimal mating time (Kemp and Soede, 1996). Therefore,

inappropriate mating time could be the reason for smaller subsequent litter size of these sows (Nissen

et al., 1997). On the other hand, Love (1979) showed that firstparity sows which mated 12 days

after weaning produced one piglet per litter more than sows with shorter weaning to conception

interval. The negative effect of heat stress on litter size in commercial herds may be confounded with

the positive effect of a delayed return to oestrus. Malovrh et al. (1996) found significant influence

of month year interaction on litter size at birth under Slovenian condition. Changes in litter size

with season were not periodical, which suggested that causes for those changes could be additional

environmental factors, such as nutrition, management etc.

2.2.4 Genotype

Genotype usually presents the effect of breed and/or crosses. In general, genotype affects litter size

because all functions of the body are under genetic control to a greater or lesser extent. It seems

obvious that all mechanisms which determine reproductive efficiency are directly influenced by ge-

netics (Clark and Leman, 1986b). Some breeds have better reproductive performance than others

(Rothschild and Bidanel, 1998; Tummaruk et al., 2000a), although differences within breed also exist

(Table 1). Some local as well as terminal breeds have usuallysmaller litter size. The largest litter size

was found in some Chinese breeds. Well known Chinese breed Meishan has on average 3 to 4 piglets

at farrowing more than commercial modern breeds (Bidanel, 1990). Crossbred sows have larger litter


size than purebred sows due to non-additive genetic effects(Rothschild and Bidanel, 1998). Johnson

(1992) found 6 to 18 % advantage due to heterozis.

Crossbreeding represents the easiest way to improve fertility. These methods are well known and nu-

merous investigations have revealed significant differences between purebred and crossbred animals.

The heterozis effect makes crossbred pigs better than the parent average. Non-additive effects result

in increased litter size of 3 to 10 %. In practice, the number of crosses to choose from is limited,

since the acceptable performance in growth, feed efficiency, and carcass traits is the most pressing

demand (Johansson, 1981). Crossbreeding experiments withhighly prolific Chinese breeds were not

successful enough. The adverse effect of Chinese breeds on growth and carcass leanness precludes

the use more than 25 % of their germplasm in commercial pig production. Young (1998) reported

that crossbred gilts containing 25 % Meishan, 25 % Fengjing,or 25 % Minzhu reach puberty earlier,

have larger litters, but they do not have any significant advantage in litter size at second parity.

dam

sow service sire

litter

sire

Figure 1: Relationship between genotypesSlika 1: Razmerje med genotipi

The effect of genotype on litter size can be observed from different points of view. Litter size is

determined by sow, sire, and piglet genotype (Figure 1). Genotype of service sire has usually small

influence on litter size as reported by Buchanan and Johnson (1984), although individual boars within

genotype could produce substantially larger or smaller litters in relation to average litter size in a

population.First of all genotype of piglets influences survival ability of piglets and therefore litter size

too. On the other hand, competition between litter mates, assome kind of interaction among piglets,

could also affect survival ability.


Table 1: Number of piglets born alive by different genotypesTabela 1: Število živorojenih pujskov pri razlicnih genotipih

Sow genotype Number of piglets born alive Source

1st parity Sows All parities

Duroc 9.2 Chen et al. (2003)

8.7 9.1 Irgang et al. (1994)

9.3 Skorupski et al. (1996)

Hampshire 9.5 Chen et al. (2003)

9.2 See et al. (1993)

Pietrain 9.2 9.8 9.7 Hamann et al. (2004)

Swedish Landrace (11) 9.6 Logar and Kovac (2001a)

9.5 Logar et al. (1999)

10.7 Tummaruk et al. (2000a)

9.9 Tummaruk et al. (2001)

German Landrace 9.8 10.5 10.4 Hamann et al. (2004)

10.4 Tölle et al. (1998)

Dutch Landrace 10.3 Hanenberg et al. (2001)

Landrace 9.7 Alfonso et al. (1997)

9.7 Babot et al. (1994)

10.4 Chen et al. (2003)

9.5 10.2 Crump et al. (1997)

9.9 Ferraz and Johnson (1993)

9.1 9.5 Irgang et al. (1994)

10.6 Marois et al. (2000)

9.5 10.2 Mercer and Crump (1990)

10.5 See et al. (1993)


8.9 Southwood and Kennedy (1990)

Large White (22) 9.5 Babot et al. (1994)

9.8 10.6 Bolet et al. (2001)

10.6 Chen et al. (2003)

7.9 9.1 Duc et al. (1998)

10.1 Ferraz and Johnson (1993)

9.2 9.7 Irgang et al. (1994)

10.0 Logar and Kovac (2001a)

9.4 Logar et al. (1999)

10.8 Marois et al. (2000)


10.2 Tölle et al. (1998)

Yorkshire 9.0 Southwood and Kennedy (1990)

10.4 Tummaruk et al. (2000a)

9.8 Tummaruk et al. (2001)

Yorkshire x Landrace 8.9 Southwood and Kennedy (1990)

Landrace x Yorkshire 8.9 Southwood and Kennedy (1990)

Hybrid 12 10.3 Logar et al. (1999)


Hybrid 21 10.2 Logar et al. (1999)



2.2.5 Direct additive genetic effect

Direct additive genetic effect is a sum of mostly small allele effects of the many genes. A half of

the alleles carried by each parent are passed from parents ontheir progeny. The measure of direct

animal genetic effect is the additive genetic effect and itsproportion (heritability) in total phenotypic

variance. Estimates of heritability (Table 2) vary mainly from 0.0 to 0.2. The general conclusion

is that the heritability of litter size is around 0.10, as reviewed by Rothschild and Bidanel (1998).

Although estimates of heritability for litter size are generally low, direct additive genetic variance was

sufficient to obtain significant genetic progress (Southwood and Kennedy, 1990; Chen et al., 2003).

Heritability estimates are slightly but not significantly higher for the number of piglets born total than

for the number of piglets born alive (Mercer and Crump, 1990;Roehe and Kennedy, 1995; Logar

et al., 1999; Hanenberg et al., 2001). Evidently there is a variation in heritability for NBA between

parities. Lower heritabilities were estimated in the first parity compared to the last parities (Roehe and

Kennedy, 1995). Alfonso et al. (1997) found high heterogeneity in heritability estimates (from 0.01 to

0.09) using univariate analyses for the first five parities inLandrace populations. Heritabilities in the

first four parities obtained by multivariate analyses were lower than the ones from univariate analyses

and ranged from 0.02 to 0.04. Heritability estimates for liveborn piglets in the first six parities using

the combination of two trait analysis varied mainly between0.05 and 0.25 in the study of Kovac

and Sadek-Pucnik (1997). Exceptionally high estimates were found in twotrait analyses for higher

parities and Kovac and Sadek-Pucnik (1997) explained them as a consequence of numerical problems.

Heritabilities were lower in the multiple trait than in the univariate analyses (Hanenberg et al., 2001).

Lower heritabilities for NBA were also found by Perez-Enciso and Gianola (1992) in Iberian pigs.

Breed differences in heritability may exist (Gu et al., 1989; Babot et al., 1994), but there is little

reliable evidence for them (See et al., 1993; Chen et al., 2003). Irgang et al. (1994) reported that

higher estimates of heritability for some breeds indicate the opportunities for genetic improvement

of litter size in these breeds that may be greater than in others. Ehlers et al. (2005) showed that

heritabilities for NBA for purebred and crossbred sows weresimilar and that pooling of data may be

considered in order to increase the accuracy of breeding value estimates. In the study from Ferraz

and Johnson (1993), two of four estimates of heritability for NBA in Yorkshire and Landrace sows, in

two herds by breed, were zero. Authors explained so low estimate of heritabilities as a consequence

of small data sets. Kisner et al. (1996) reported estimates of heritabilities in the first three parities in

a wide range from zero to 0.10 and explained them also as a result of small number of records in the

data set, as well as data structure.

Several reasons for the heterogeneity of heritability estimates were reported by Southwood and

Kennedy (1990). The main reasons were random estimation error, breed and time at which litter

size is measured, unaccounted genetic and environmental sources. Southwood and Kennedy (1990)

also found that heritability estimates were reduced considerably by not accounting for maternal effect

and its correlation with direct additive genetic effect, but only for one of two breeds in the study.

However, the results of their study were not consistent withthe findings of Crump et al. (1997) who

observed small changes when maternal genetic effect was included in the model. Heritability esti-


Table 2: Heritability estimates for number of piglets born alive from the literatureTabela 2: Ocene heritabilitete za število živorojenih pujskov iz literature

Author No. of

records

Parity Genotype h2

a h2

m ram Random

effects

Southwood and Kennedy (1990) 4225 1 LR 0.09 a

2960 1 LW 0.13 a

Irgang et al. (1994) 5799 1 LR 0.14 a

3576 2 LR 0.20 a

2356 3 LR 0.02 a

4561* 1 LW 0.09 a

2862 2 LW 0.15 a

2004 3 LW 0.17 a

Duc et al. (1998) 2035 1 LW 0.06 a

1802 2 LW 0.09 a

1488 3 LW 0.10 a

1225 4 LW 0.08 a

Hermesch et al. (2000) 5986 1 LR, LW 0.08 a

4113 2 LR, LW 0.09 a

2965 3 LR, LW 0.08 a

Alfonso et al. (1994) 27055 1-5 LW x SL 0.05 a, p

1 LW x SL 0.02 a

2 LW x SL 0.02 a

3 LW x SL 0.02 a

4 LW x SL 0.04 a

5 LW x SL 0.04 a

Sorensen (1990) ? 1-11 LW 0.12 a, p

Babot et al. (1994) 13092 1-9 LR 0.05 a, p

2760 1-9 LW 0.09 a, p

Alfonso et al. (1997) 34417 1-5 LR 0.05 a, p

34417 1-5 LR 0.05 <0.005 -0.03 a, p, m

Adamec and Johnson (1997) 12077 1-17 LW, SL 0.10 a, p

12077 1-17 LW, SL 0.12 0.000 -1.00 a, p, m

Roehe and Kennedy (1995) 11782 1 LW 0.11 0.010 -0.29 a, m

8084 2 LW 0.07 0.030 -0.09 a, m

5904 3 LW 0.11 0.003 −b a, m

4587 4 LW 0.13 0.001 − a, m

16306 1 LR 0.11 0.003 − a, m

11120 2 LR 0.10 0.006 − a, m

8301 3 LR 0.11 0.000 − a, m

6314 4 LR 0.14 0.007 − a, m

Crump et al. (1997) 5291 1-11 LR 0.10 a, p

5291 1-11 LR 0.09 a, p, c

5291 1-11 LR 0.09 0.010 -0.09 a, p, m

Chen et al. (2003) 251296 ? LW 0.10 0.010 -0.27 a, p, m, s

53224 ? LR 0.07 0.020 -0.70 a, p, m, s

LW - Large White; LR - Landrace; SL - Swedish Landrace; a - direct additive genetic effect; p - permanent environment effect; m -

maternal additive genetic effect; c - common litter environmental effect; s - service sire effect; ? - not provided; * number of sows;h2a-

direct heritability; h2m- maternal heritability;ram- correlation between direct and maternal additive geneticeffect; bCorrelations were

only presented when the maternal heritability is≥1 %.


mates are also affected by the method of estimation (Holl andRobison, 2002). It seems that estimates

obtained by fitting an animal model were in the narrower range(from 0.09 to 0.14) than estimates

obtained from older methods like daughter-dam regressionsor half-sib analyses. Relatively high es-

timate of 0.22 for heritability obtained using REML was found by Kaufmann et al. (2000). Estimates

of heritabilities are a function of variance components andare, in general, specific for a particular

population and period of time (Kaplon et al., 1991).

2.2.6 Maternal additive genetic effect

Maternal effect indicates that dam has an influence on the performance of her offspring. The maternal

effect of a sow is a function of both her genotype and environment. Intrauterine environment, milk

production, and mothering ability of the dam may affect her offspring’s reproductive performance.

Although maternal effects are strictly environmental withrespect to offspring, these effects can have

both environmental and genetic components. In practice, the maternal environmental effect would be

accounted for by the common environmental effect of birth ofa sow (Roehe and Kennedy, 1993b).

The presence of maternal additive genetic effect may bias estimates of direct additive genetic effects

because both are transmitted from one generation to the next. Genetic differences among dams are

expressed as phenotypic differences of their offspring when they become dams, i.e., the additive

maternal effect is expressed one generation later than the additive direct effect. Estimates of maternal

genetic effect were low and ranged from 0.00 to 0.05 (Chen et al., 2003).

Genetic progress for litter size may be reduced because of negative correlation between direct addi-

tive genetic and maternal additive genetic effects. It has been suggested that an antagonistic genetic

correlation between direct and maternal genetic effects may be responsible for the observed low her-

itability and lack of response to selection for litter size (Southwood and Kennedy, 1990). Simulation

study by Southwood and Kennedy (1990) showed that maternal additive genetic effect with negative

correlation to direct additive effect from the first parity litters had a considerable effect on response to

selection of litter size. A great advantage of an animal model is that it provides unbiased estimates for

maternal and direct effects of all animals with and without records. This is important because the esti-

mation of maternal effect for gilts and young sires is based on records that are at least one generation

behind those of the direct effect (Roehe and Kennedy, 1993a). Therefore, if maternal genetic effect

is important, long-term breeding objective should be implemented by a complete animal model.

The importance of maternal additive genetic effect has beencontroversial. Using REML under an

animal model, Mercer and Crump (1990) and Perez-Enciso and Gianola (1992) found no maternal

effect, whereas Southwood and Kennedy (1990) and Ferraz andJohnson (1993) reported significant

maternal effect. Ferraz and Johnson (1993) compared modelswithout and with maternal additive ge-

netic effect and tested them using likelihood ratio test. Low maternal heritability could be explained

by the large amount of crossfostering practice, often within 24 hours from birth, which means that

sows who were litter mates at birth did not necessarily sharethe same postnatal environment (Crump

et al., 1997). A negative genetic correlation between the direct and maternal effect could lead to con-


flict in the improvement of a trait. Although the maternal environmental component may be removed

by appropriate management or statistical methods, such as crossfostering or adjustment of the data,

the genetic part may not be accounted for. Roehe and Kennedy (1993a) noticed that maternal genetic

effect can have a high influence on genetic improvement of litter size, even when maternal heritabil-

ity is low compared to direct heritability, depending on thegenetic correlation between maternal and

direct genetic effects.

Various reasons for contradictory results for maternal effect are possible. Different populations, dif-

ferent environmental conditions, crossfostering etc., are the most frequent reasons listed. Different

models and traits used could also have an influence on the estimates of the maternal effects. Van-

gen (1981) showed that maternal effect mainly affects the first litter. Therefore, maternal effect,

if important in the population, should be included in the evaluation model for the first parity litter

(Roehe and Kennedy, 1993b). In subsequent parities, the statistical model may be different because

of the less important maternal effect. See et al. (1993) suggested that maternal genetic effect might

play a different role within different breeds. Ignoring maternal effect in a selection program for sow

productivity could actually prevent genetic progress (Southwood and Kennedy, 1990). Even if ge-

netically uncorrelated, direct effect would be biased because maternal effects are embedded in the

phenotype expression of litter size and inherited over the same genetic paths as direct effects (Roehe

and Kennedy, 1993b). Genetic trend may be overestimated with the model without maternal effect.

As a result, younger animals with lower true genetic merit would be more frequently selected than

older animals with higher genetic merit (Roehe and Kennedy,1993b).

The maternal effect should not be seen just as a nuisance effect that impedes the improvement of

direct genetic effect. The maternal response to selection may be important. This may be particularly

true for crosses between prolific Chinese breeds and modern breeds. Roehe and Kennedy (1993a)

reported that with a large negative correlation between maternal and direct genetic effect an acceptable

response to selection can be obtained only with a crossbreeding scheme. Equal weighting of maternal

and direct genetic effects in the female dam line was superior in the production of crossbred sows

when both effects were independent. But, Roehe and Kennedy (1993a) explained that when the

genetic correlation between maternal and direct effect wasvery large (-0.9), substantially greater

response was achieved when the female dam line was selected with the weighting ratio 4:1 of maternal

and direct effects. One disadvantage of selecting for a highmaternal response was a negative overall

response in the purebred line.

2.2.7 Sire effect

Sire influences litter size directly and indirectly. Service sire as a father of the litter affects litter size

directly through semen characteristics (quality and quantity) and through genetic potential for devel-

opment and survival of embryos. Sire, as a father of the sow, affects litter size indirectly through

fertility of his daughters which is described by direct additive genetic effect. It is necessary to dis-

tinguish service sire of the litter and sire of the sow. Although sire effect presents mainly a genetic


effect, it can also include environmental components. Different semen dilution as well as different

interactions between animal and human at mating time can be the reason for differences in litter size.

With a reported estimate up to 5 % of the total variation in thenumber of piglets born alive, it seems

that the service sire has a small but significant effect on thenumber of piglets born alive (See et al.,

1993; Hamann et al., 2004).

Service sires within genotype could produce substantiallylarger or smaller litters in relation to average

litter size in a population. Sires can produce small littersif semen concentration and quality are so low

that not all eggs are fertilized (Clark and Leman, 1986b). Individual sires can produce small litters if

lethal genes are produced that result in death of a portion ofembryos (Ollivier, 1982). Chromosomal

aberrations have been found to reduce fertility in sires (Locniškar, 1974; Popescu, 2004). Litter size

is then typically reduced by 50 to 75 % and more due to an increased embryonic mortality. Estimates

of the contribution of the sire of a litter to the variance in litter size are generally small, and seem

dependent on whether service is natural or by artificial insemination (Strang, 1970; Haley et al.,

1988). Southwood and Kennedy (1990) assumed that service sire did not have an important role on

litter size. The effect, which may be seen as ’fetal effect’ on sow prolificacy, explains up to 5 % of the

total variance in litter size (Ollivier, 1982). However, Ollivier (1982) argued that service sire effect

is more important than those explained by the genetic effecttransmitted by the sire to his daughters.

Therefore, a possibility exists, to maintain litter size ata satisfactory level by culling the less prolific

sires.

2.2.8 Common litter environmental effect

Common litter environmental effect describes common environment shared by sows from farrowing

to weaning, and in some cases, growth period if animals continue to stay together after weaning. Be-

cause of common environment, resemblance between litter mates as well as diversity among different

litters increase. Variance of common environment is causedby microclimate, milk production, other

maternal ability, nutrition, hygiene and so on. Although the effect presents the usual environmental

component, common litter effect contains some genetic components like dominance and maternal

genetic effects, if they are not included in the model (Boletet al., 2001). Estimates of common litter

environmental effect were low and ranged usually between 0 and 6 %. Crump et al. (1997) consid-

ered that lower estimates of the effect may be a consequence of the routine practice of crossfostering.

Kaplon et al. (1991) found higher estimates in range between0.03 and 0.11. Kaufmann et al. (2000)

reported that common litter environmental effect explains6 % of phenotypic variance. Small magni-

tude is often described as a consequence of a small number of litter mates in the data. If more than

70 % of the records presented sows with no litter mate or the average number of sows per litter was

below 1.5, Southwood and Kennedy (1990) suggested that the effect could be assumed to be unim-

portant and thus, not included in the model. Hermesch et al. (2000) agreed that data containing less

than 20 % of sows with one full sister should be analysed to fit litter effect as an additional random

effect. Adding common litter effect to the animal model did not cause any sizable reduction in the

residual variances or change of heritabilities (Irgang et al., 1994).


2.2.9 Permanent environmental effect

Permanent environmental effect is connected with more measurements on the same animal. Because

of the same permanent environment, measurements obtained on the same animal are more similar

than measurements obtained on the others. Frequently, estimates ranged between 5 and 10 %. Al-

fonso et al. (1997) and Hanenberg et al. (2001) estimated that 7 to 9 % of phenotypic variance was

explained by permanent environmental effect. In both casesrandom part comprised only direct ad-

ditive genetic and permanent environmental effects. Chen et al. (2003) considered additional service

sire and maternal additive genetic effect in the model and obtained estimates for permanent environ-

mental effect between 0.06 and 0.08 for four different breeds. The importance of including this effect

in the model was given by Ferraz and Johnson (1993), which reported estimates of permanent envi-

ronmental effect as a ratio in phenotypic variance in the range between 0.16 and 0.17. On the other

hand, Adamec and Johnson (1997) found lower estimates of permanent environmental effect, in the

range between 0.02 and 0.03.

2.3 GENETIC CORRELATIONS

Biological reasons for the change of genetic correlations between litter size in different parities could

be mainly related to the maturation of animals. Namely, the sows give the first litters at the age when

they are still in the growing stage, when genetic potential for production is not wholly achieved yet.

Further, a difference in genetic correlations could be the consequence of culling and smaller number

of data in later parities, as well as a problem of optimal mating time in gilts. The size of genetic

correlations between parities is important to define optimal evaluation procedures to improve litter

size in pigs. Genetic correlations between parities of one,or approximately one, indicate that genetic

gain in gilts would assure genetic gain in later parities, with the benefits of reduced generation interval

and increased selection intensity. High genetic correlations indicate strong genetic ties between litter

size in gilts and sows. Thus, litter size across parities could be treated as the same trait. Many breeding

programmes applied repeatability model for the estimationof breeding values for litter size.

Low genetic correlations indicate that litters from different parities should be treated as different

traits, which can be modelled by multiple-trait animal models for genetic evaluation and estimation of

genetic parameters. Estimates of genetic correlations forlitter size between parities varied from 0.37

to 0.99 (Table 3). Although Avalos and Smith (1987) and Haleyet al. (1988) suggested analyzing

litter size as repeated records, results from more recent studies (Alfonso et al., 1994; Irgang et al.,

1994; Roehe and Kennedy, 1995; Rydhmer et al., 1995; Hanenberg et al., 2001) confirmed that the

number of piglets born alive in the first parity should be regarded as a different trait than litter size

in later parities. More recently, in Bayesian analysis of litter size for multiple parities, heterogeneous

genetic response across parities have suggested that litter size in each parity may have a different

genetic background (Noguera et al., 2002). Johansson (1981) and Irgang et al. (1994) reported a

genetic correlation around 0.40 between the first and the second parity for the number of piglets born

alive. Similarly, Duc et al. (1998) cited that genetic correlations between the first and subsequent


Table 3: Genetic correlation for the number of piglets born alive between successive paritiesTabela 3: Primerjava genetskih korelacij za število živorojenih pujskov med zaporednimi prasitvami

Author Parities Analysis Genetic correlationsIrgang et al. (1994) 1, 2 MT 0.37 - 0.48

1, 3 0.76 - 0.992, 3 0.77 - 0.99

Rydhmer et al. (1995) 1,2 TT 0.70Roehe and Kennedy (1995) 1, 2-4 MT 0.87 - 0.49

2, 3-4 0.79 - 0.66Alfonso et al. (1997) 1, 2-5 MT 0.74 - 0.57Duc et al. (1998) 1, 2-4 MT 0.87 - 0.41

2, 3 0.87 - 0.97Tölle et al. (1998) 1,2 MT 0.90

1,3 0.982,3 0.95

Hermesch et al. (2000) 1-3 MT 0.52 - 0.78Logar and Kovac (2001a) 1, 2-6 TT 0.92Serenius et al. (2003) 1, 2-4 MT 0.77 - 0.32

2, 3-4 0.96 - 0.59

MT - multiple-trait; TT - two trait

parities (0.41-0.87) were lower than between the second andthe adjacent parities (0.87-0.97). Low

genetic correlations may indicate that partly different genes are responsible for litter size in different

parities. However, Johansson (1981) reported that it is most likely that lower correlations arise only

between the first and other litters. High genetic correlations among litter size in latter parities were

connected with numerical problems in multiple-trait analysis (Sadek-Pucnik and Kovac, 1996). High

genetic correlation (0.92) between the number of piglets born alive in gilts and sows was obtained

in the study from Logar and Kovac (2001a). In addition to the genetic correlations, two different

aspects should be considered when repeatability or multivariate analyses are discussed for practical

purposes. The routine genetic evaluation will be considerably more costly in computer time with the

multivariate model. Secondly, herd-year-season or other contemporary groups are smaller whenever

litter size is considered as a different trait per parity. Estimates of such contemporary groups will be

less accurate and the accuracy of estimated breeding valueswill decrease (Hanenberg et al., 2001).

The knowledge about genetic correlations between litter size and other reproduction or performance

traits is very important for the optimal selection strategy. This is required in order to establish whether

reproductive traits should be analysed in a multivariate analysis. Genetic correlations between the

number of piglets born total and the number of piglets born alive were high (over 0.90). Thus,

selection for these two traits will result in similar response in each trait (Roehe and Kennedy, 1995;

Logar et al., 1999). However, selection for the number of piglets born total impose a risk of increasing

the number of piglets born dead. Selection for the number of piglets at weaning is very difficult under


conditions of crossfostering. Genetic correlations between the number of piglets weaned and litter

size at birth were moderate. Litter size at weaning is of greater commercial importance than litter size

at birth. Therefore, management to prevent crossfosteringmay be desirable in nucleus herds to obtain

a more accurate genetic merit for litter size at weaning.

2.4 STATISTICAL MODELS FOR GENETIC EVALUATION OF LITTER SIZE

From statistical point of view, effects can be classified as fixed or random (Searle et al., 1992). The

effects are considered as fixed whenever they can be attributed to a finite set of levels in the data

and which are there because we are interested in them. For fixed effects, there must be usually

enough information in the data. Random effects are attributable to a (usually) infinite set of levels,

of which only a random sample is considered to be included in the data. Additionally, random

effects usually have a small number of observations per level and at the same time a large number of

levels. The structure of random effects is presented on Figure 2. Random effects could be genetic or

environmental. In most animal breeding applications, onlyadditive genetic effects are considered in

the evaluation of animals. Beside additive genetic models,there is an increased interest in models

that consider non additive genetic effects, as dominance. Pigs, as species with a large number of

dominance relationships (full sibs) and populations that use specialized sire and dam lines to im-

prove reproductive traits, seem well suited for dominance model. The results from Culbertson et al.

(1998) indicated that dominance effect may be important forlitter size. The sows influence on the

performance of her offspring is not limited only to genetic effects, but also includes environmental

random effects as permanent and common litter effects. The knowledge of variances and covariances

of random effects that affect litter size is necessary to define how breeding values should be estimated.

EFFECTSRANDOM

GENETIC

ADDITIVE

PATERNAL

MATERNAL

DIRECT

NON−ADDITIVE (DOMINANCE)

PATERNAL

INDIVIDUAL

ENVIRONMENTAL

COMMON

LITTER

PERMANENT

TEMPORARY

Figure 2: Structure of random effectsSlika 2: Struktura nakljucnih vplivov

Some effects could be in one situation treated as fixed or random in another, depending on the number

of levels and the number of observations per level, as it is often the case with a herd or contemporary


group effect. In order to avoid bias, all significant effectsshould be included in the models for genetic

evaluation.

Determination of fixed effects in a model is one of the first steps in model development of genetic

evaluation of animals. Fixed part of the model for litter size usually explains a small part of the total

variability. The use of the incorrect models will lead to predictors which are neither optimal nor

unbiased (Babot et al., 1994). On the other hand, including effects of little relevance increases the

variance of prediction error and thus reduces the accuracy of estimated breeding value. At the same

time, the computational costs increase, although the laterremark is not so important considering the

progress of computer power.

An overview of the fixed part of models applied for in genetic evaluation for litter size is given in

Table 4. The first group of effects consists of explanatory variables describing age of the sow at far-

rowing or parts of reproduction cycles that influence littersize in subsequent litters. The next group

is used to describe seasons or contemporary groups. Further, it is followed by a group that incorpo-

rates the genetic components (sow and sire genotype, as wellas service sire). The last group contains

effects connected with mating management, like mating typeand the number of inseminations. Age

and reproduction cycle variables were almost always presented as regressions. All other effects are

presented as class effects. Effects that are most often included in statistical models for litter size are

parity and age at farrowing.

Parity effect is presented as class effect. Most studies include up to six, and seldom more than ten

parities. Usually, if data contain later parities as well, the last parities are combined in one joined

class (Babot et al., 1994). The reason is a probably small number of records in later parities. The

first two parities are almost always defined as separate classes, while later parities are combined

into classes (Alfonso et al., 1997). The effect of age at farrowing on litter size is presented by a

few different ways. Firstly, there is a difference in approach where some authors show only the

effect of age at first farrowing on litter size. Secondly, theeffect of age at farrowing on litter size is

described using different regressions. Most often it is presented with linear (Duc et al., 1998; Tribout

et al., 1998; Marois et al., 2000; Hermesch et al., 2000) or quadratic regression (Roehe and Kennedy,

1995; Kovac and Sadek-Pucnik, 1997; Logar, 2000). Logar (2000) reported that quadratic regression

efficiently described a change of litter size by age at first farrowing, while the number of piglets born

alive increased up to the age of one year. Very rarely higher polynomials as cubic regression were

used to describe the effect of age at farrowing (Frey et al., 1997). In the studies by Tribout et al.

(1998) and Marois et al. (2000) the age at farrowing was used for all parities. By nesting them within

parities, they obtained regression curve for each parity separately.

Lukovic

Z.C

ovariancefunctions

forlitter

sizein

pigsusing

arandom

regressionm

odel.D

octoralDissertation.

Ljubljana,U

niv.ofL

jubljana,Bio

technicalFaculty,Z

ootechnicalDepartm

ent,200620

Table 4: Fixed effects in the models for the number of pigletsborn aliveTabela 4: Pregled sistematskih vplivov v modelih za številoživorojenih pujskov

Author/Effect

Par

ity

Age

atfir

stfa

rrow

ing

Age

atfa

rrow

ing

Far

row

ing

inte

rval

Lac

tatio

nle

ngth

Wea

ning

toco

ncep

tion

inte

rval

Her

d

Yea

rof

farr

owin

g

Mon

thof

farr

owin

g

Her

dye

arse

ason

Her

d-ye

arof

farr

owin

g

Sea

son

offa

rrow

ing

Mat

ing

seas

on

Sow

geno

type

Ser

vice

sire

geno

type

Ser

vice

sire

Mat

ing

type

Num

ber

ofin

sem

inat

ions

Adamec and Johnson (1997) + + + + +

Alfonso et al. (1997) + +b

Babot et al. (1994) +a LR LR +

b

Duc et al. (1998) + LR +b + +

Frey et al. (1997) CR + + +

Hamann et al. (2004) + +b

Hanenberg et al. (2001) + +c + +

d

Hermesch et al. (2000) LR +b + +

Irgang et al. (1994) LR +b

Kovac and Sadek-Pucnik (1997) QR LR +e +

Logar and Kovac (2001a) + QR LR LR +e +

Marois et al. (2000) + LR LRi LR + +f

Perez-Enciso and Gianola (1992) + +

Roehe and Kennedy (1995) LR,QR LR,QR + +g + +

d

Tölle et al. (1998) LR + + +

Tribout et al. (1998) + LRi + +

anine classes: 1-8, >8;bthree months interval;cseason in months;done or more than one;eyear-month interaction;f three classes: 1, 2, >2;gfour months interval;inested within parity;

LR - linear regression; QR - quadratic regression; CR - cubicregression


In sows, different authors tried to adjust litter size with the length of farrowing interval or their compo-

nents like lactation or weaning to conception interval. Theeffect of previous lactation length on litter

size in statistical models was mainly presented as linear (Kovac et al., 1983; Xue et al., 1993; Logar,

2000; Marois et al., 2000) and very rarely as quadratic regression (Marois et al., 2000). Regarding

lactation length, studies differed in the length of interval observed. In some studies, short lactations

with higher decrease of litter size, were excluded from dataset. Weaning to conception interval was

mainly presented as linear regression. However, linear regression did not modell changes in litter

size well when insemination occurred 7 days or more after weaning. In the recent study by Marois

et al. (2000), it is shown that the effect is curvilinear and aspecific decrease in litter size cannot be

presented as linear or quadratic regression.

Contemporary group effect showed the largest difference indefinition of the effect. Related to avail-

able data, different combinations of season and some other effects, as herd or region were used (Frey

et al., 1997). Season effect is usually defined as month, three month or four month interval. Most

frequently the season of mating and farrowing were included.

Genotype of sow or sire, as well as service sire were very rarely included in models for litter size.

Genotype of sow presented mainly the purebred sow (Adamec and Johnson, 1997; Hermesch et al.,

2000), and very rarely crosses. Similar situation was in sire genotype too. In some studies where

different types of mating were provided, the effect was usedin two levels; artificial insemination or

natural service. The number of inseminations was used sometimes, with maximum three levels: one,

two, and more than two inseminations. The effects related tomating management were included in

the model rarely due to the absence of recording.

Genetic progress in animal breeding depends on accurate estimates of variances and heritabilities for

traits. The knowledge of genetic parameters for litter sizeis necessary to estimate breeding values

accurately, to optimize breeding schemes, and to predict genetic response due to selection (Roehe

and Kennedy, 1995). At present, mixed model methodology is the method of choice for genetic

evaluation. The methodology consists of a framework with justifiable statistical and genetic properties

and it potentially delivers the best unbiased predictors ofbreeding values. The quality of evaluations

depend mainly on the data and the model. The choice of model for genetic evaluation depends on

genetic correlations between litter size in different parities, size of data, number of parities included,

computer capacity.

For traits like litter size which repeat over time, the general question is whether to use a repeatability

model or a multiple-trait model. Many breeding programmes use repeatability model for the estima-

tion of breeding values for litter size due to its simplicity. Alfonso et al. (1997) reported that most

criteria showed that the use of the repeatability model was more appropriate and only the pessimistic

criteria suggested the use of a multiple-trait model for thegenetic evaluation of litter size. The use of

a multivariate evaluation for litter size is not clearly justified, due to the fact that the results obtained

under a multiple-trait model are not reliable enough and those obtained under bivariate models are not

sufficient to recommend a multiple-trait animal evaluationmodel with litter size in different parities

treated as different traits. In practice, the choice of the model does not only depend on the genetic


correlations but also on problems related to data structure. Namely, treating the first and the second

parities as separate traits often reduces the size of herd-year-season subclasses. This can cause mas-

sive problems in the data structure that reduce the accuracyof estimated breeding values (Götz, 1998;

Hanenberg et al., 2001). In general, the estimation of genetic parameters using multivariate model

will be considerably more costly in computer time.

2.4.1 Repeatability model

Repeatability models treat litter size at successive parities as repeated measurements of the same

trait. It has been used frequently in the past because of its simplicity (Noguera et al., 1998) and small

computer capacity. With several measurements per animal, it requires relatively little computational

efforts. The repeatability model incorporates genetic correlations of one among subsequent observa-

tions and a constant genetic variance along trajectory. Litter size in different parities is considered as

the result of the same group of genes during the female’s life. With such approach, only one breeding

value is predicted for all records included in genetic evaluation. Validity of repeatability model for

genetic evaluation of litter size was often checked. The useof two different models to adjust the

first parity for age at first farrowing and later parities for lactation length and weaning to conception

interval was the main reason for not using a simple repeatability model (Logar et al., 1999). A way

how to handle repeated records with different fixed effects was presented by Andersen (1998).

2.4.2 Multiple-trait model

Multiple-trait analysis treats subsequent observations to be different traits. Estimates of genetic corre-

lations between litter size in different parities were sometimes substantially lower than one (Alfonso

et al., 1994; Irgang et al., 1994; Roehe and Kennedy, 1995), especially between the first and later

parities (Johansson, 1981). Low correlations between litter size in different parities may indicate that

partly different genes are responsible for litter size in different parities. Estimates of genetic parame-

ters for litter size can be biased by involuntary culling from parity to parity. One approach to account

for this selection bias is to treat different parities as different traits (Roehe and Kennedy, 1995). Con-

sidering litter size in each parity as a separate trait implies the estimation of (co)variances between the

traits. A certain approach can be used to examine whether litter size in different parities is genetically

the same trait. Therefore, multiple-trait analysis is preferred in such a situation in order to increase

the efficiency of selection for litter size. In the study by Serenius et al. (2003) litter size was modelled

by multiple-trait model rather than repeatability model when genetic correlations between liter sizes

in different parities were low. Although a univariate modelcould suffice in genetic evaluations, the

use of a multivariate model could be interesting in order to avoid overestimation of expected selection

response. Alfonso et al. (1994) reported that the use of a univariate animal model implies a 14 % of

loss in expected response in relation to the multivariate animal model. One of the primary reasons

for multiple-trait analyses is to increase the accuracy of the evaluations, especially for traits with low


heritability. However, when correlations among traits arelow, multiple-trait analyses will have little

impact on the accuracy of breeding values for the low heritable traits (Short et al., 1994).

Multivariate analyses were not always successful (Alfonsoet al., 1997). With the assumption that lit-

ter size in different parities is a different trait, multiple-trait model requires more computational efforts

and more parameters to be estimated than a repeatability model. Algorithms based on the likelihood

function do not guarantee accurate solutions in multiple-trait models with numerous components to

be estimated (Groeneveld and Kovac, 1990). Two main reasons reported by Misztal (1994) could

explain this. Firstly, accuracy of likelihood computationdecreases if traits are highly correlated, and

secondly, maximisation methods may fail when the likelihood function is becoming flatter due to a

high number of parameters to be estimated.

2.4.3 Random regression model

More recent approach has been proposed for longitudinal data (Schaeffer and Dekkers, 1994).

Multiple-trait model was replaced by random regression models. The main advantages of RRM

approach in comparison to multiple-trait models are: smaller number of parameters to describe longi-

tudinal measurements, smoother (co)variance estimates, as well as a possibility to estimate covariance

components and predict breeding values at any point along the trajectory. By fitting a set of random

regression coefficients, production functions are modelling genetic and environmental changes over

time (Meyer, 1998). Regression coefficients are generally treated as fixed to account for overall trends

within some fixed effects. However, they can be fitted as a set of regression coefficients within ran-

dom effects to describe specific production curve. Such regression coefficients are allowed to vary

according to the distribution assigned to them. Therefore they are indicated as random regression

coefficients. The development of statistical models with random regression coefficients enable mod-

elling of production curves as a function of age or some othertime or space variable for individual

animals. Models split the production curve into fixed and random parts. Fixed part describes a gen-

eral shape of production curves common to the whole population or certain contemporary groups.

Random part covers specific deviations of the individual production curve from its common shape

defined in a fixed part.

Random regression models have a basic structure that is similar in most applications. According to

Schaeffer (2004) a simplified RRM for a single trait can be written as

yijkn:t = Fi + g(t)j + r(a, x,m1)k + r(pe, x,m2)k + eijkn:t ... (1)

Dependent variableyijkn:t is thenth observation on thekth animal at timet belonging to the time

independent (Fi) and thejth group for time dependent (g(t)j ) fixed effects;g(t)j term denotes one

or more functions that account for the phenotypic trajectory of the average observations across all

animals belonging to thejth group;r(a, x,m1)k andr(pe, x,m2)k are notations adopted for random

regression functions, wherea denotes direct additive genetic effect of thekth animal,pe denotes


permanent environmental effect of thekth animal,x is the vector of time variables,m1 andm2 denote

orders of the regression functions, andeijkn:t is random residual.

Fitting the random regression model, a certain covariance structure among the observations is as-

sumed. This is determined by the covariances among the regression coefficients and can be char-

acterised by a covariance function. A covariance function can be defined as “a continuous function

to give the variance and covariance of traits measured at different points on a trajectory”. They can

be used to describe the phenotypic covariance structure, asa sum of covariance functions for all

random effects which explain the variation. In essence, a covariance function is merely the infinite-

dimensional equivalent to a covariance matrix for a given number of records taken at different ages.

It gives the covariance between any two records measured at given ages as a function of the ages

and some coefficients (Meyer and Hill, 1997). Kirkpatrick etal. (1990) and Kirkpatrick et al. (1994)

showed how the patterns of phenotypic and genetic variance can be modelled as a function of time.

Covariance functions can be written as random regression where independent variables are standard-

ised expressions of time, as shown by Meyer and Hill (1997).

Random regression has already been applied for routine genetic evaluation for milk traits in dairy

cattle (Schaeffer and Dekkers, 1994; Schaeffer et al., 2000), while it allows genetic evaluation for

the production level in the entire lactation, in a part of lactation, or on a certain day. The possibility

to predict genetic merit for the persistency in milk traits provides an opportunity to make selection

on such traits, too. Random regression for growth traits hasbeen used mainly to estimate dispersion

parameters in pigs, cattle, and sheep. In pigs, RRMs were mainly used for feed intake (Huisman et al.,

2002) and growth (Andersen and Pedersen, 1996; Malovrh, 2003). Key issues in the application

of RRM to growth traits are the frequency and distribution ofmeasurements along the observed

trajectory. Comparing to milk and growth traits, litter size in pigs has smaller changes of phenotypic

variance, and it is not a real infinite dimensional trait. Although litter size as a discrete trait differs

from the milk and growth traits, RRM can be applied for the estimation of genetic parameters for litter

size in pigs, as suggested by Schaeffer (2004). In all applications of RRM, orthogonal polynomials

are the most appropriate for the covariates. Further research is required to determine the best order of

polynomials to include in different applications.

Random regression models are intended for use on longitudinal data or repeated measurements where

observations for a trait are collected several times duringanimal’s life. Litter size in pigs is measured

more than once in sow lifetime and can be considered as a longitudinal trait too. The common

shape of litter size curve in pigs is known, and it is also known that factors such as parity, age at

farrowing, service sire, genotype of sow, etc., affect it. The effects mentioned above have sufficiently

large number of observations, which belong to the large groups of animals and therefore, they can

be estimated accurately and treated as fixed effects. The characteristics of each animal contribute

considerable to the course of common litter size curve. However, there is not enough information

about the shape of litter size curves for individual animals, and regression coefficients for individual

litter size curves can only predict differences between animals. Therefore they are treated as random.


3 MATERIAL AND METHODS

3.1 MATERIAL

Data were supplied by the Slovenian pig breeding organization. Litter records from three large farms

(A, B, C) were used. Litters were collected from sows that farrowed between January 1990 and

December 2002 for farms B and C, and from January 1992 to December 2002 for farm A. Litter

records from four sow genotype were provided. Two data sets per farm were prepared. The first data

set (DS1) contained litter records from the first to the sixthparity. The first data set was prepared

for the comparison between multiple-trait and random regression models. The second data set (DS2)

was prepared in order to verify the possibility for selection on persistency for litter size in pigs and

included litter records up to the tenth parity. Litter records were analyzed separately for each farm

due to absence of genetic connections among them.

Information of individual records could be divided into three parts. The first part included some basic

information about the litter: identification number of a sow, sow genotype, common litter, mating

season, service sire, service sire breed, dam, and parity. Common litter environment described an

environment specific for litter mates. It was denoted by identification of the litter the sow was born

in. The second part of litter record presented different measurements of litter size: number of piglets

born and number of piglets born alive. The third part of litter records included age at farrowing

and variables that described different phases in reproductive cycle: lactation, weaning to conception

interval, and farrowing interval.

Litter records were excluded from the analysis if data were outside determined range. Thus, the

records with lactation longer than 60 days or previous weaning to conception interval longer than

80 days were deleted. Another restriction was imposed for the age at farrowing. Those records with

the age at farrowing outside limits determined for each parity (Table 5) were deleted. The ranges were

defined by the examination of data in order to exclude the extreme values with too little information.

This resulted in deletion of 939 records (1.3 %) for farm A, 3220 records (2.5 %) for farm B, and

4166 records (4.1 %) for farm C, in the data sets DS2.

Table 5: Elimination criteria for age at farrowing by parityTabela 5: Kriterij za izlocitev podatkov zaradi starosti ob prasitvi po zaporednih prasitvah

Parity Age at farrowing Parity Age at farrowing1 290 - 480 6 1020 - 13002 420 - 660 7 1050 - 16003 580 - 820 8 1200 - 18504 720 - 980 9 1350 - 20005 860 - 1140 10 1400 - 2150

All records with the number of piglets born alive in the rangebetween 0 and 22 liveborn piglets were

used in the analyses. Service sires with less than ten litters were combined in groups by genotype.


After data editing, a total of 61415 litters from farm A, 99512 litters from farm B, and 85986 litters

from farm C were used in the final data sets DS1 (Table 6). A total of 70033 litters from farm A,

118079 litters from farm B, and 99411 litters from farm C wereincluded in data sets DS2 (Table 6).

The additional amount of data from the sixth to the tenth parity presented 12.30 % records on farm

A, 15.72 % records on farm B, and 13.50 % records on farm C, fromthe total number of records

included in data sets DS2.

Table 6: Data structure for data sets DS1 and DS2 for farms A, B, and CTabela 6: Struktura podatkov za niza podatkov DS1 in DS2 po farmah

FarmA B C

Characteristics DS1 DS2 DS1 DS2 DS1 DS2

Number of litters 61415 70033 99512 118079 85986 99411Number of sows 19170 19170 28364 28364 27912 27912Litters per sow 3.20 3.65 3.50 4.16 3.08 3.56Litters per service sire 124.57 93.50 109.00 128.20 139.58 160.34Sows per common litter 1.40 1.40 1.47 1.46 1.50 1.49

Data structure was similar for both data sets (Table 6). The number of litters per sow ranged from

one to ten with the average of 3.08 to 4.16 litters per sow. Data sets DS2 had expectedly more litter

records. The average number of litters per service sire ranged between 93 on farm A and 160 on farm

C in DS2. Additional litter records in data sets DS2 did not change the number of sows that shared

common litter environment. The average number of sows per common litter varied among farms

from 1.40 to 1.50. Data structure allowed the inclusion of common litter environmental effect in the

models. Between 65 and 70 % of common litters had only one sow,between 20 and 25 % of common

litters had two sows (Figure 3). Three or more sows came from around 10 % of common litters.


0

10

20

30

40

50

60

70

80

1 2 3 4 5 6 7

ABC

%

No. of sows from the same litter

Farm:

Figure 3: Relative frequency of sows per common litter by farmSlika 3: Relativna frekvenca svinj na gnezdo po farmah

Table 7: Number of animals, mean (x) and standard deviation (σ) for number of piglets born alive(NBA), age at farrowing (AF), previous lactation length (PLL) and mean and mode for weaning toconception interval (WCI) in data sets DS1 and DS2 within farmsTabela 7: Število živali, povprecje in standardni odklon za število živorojenih pujskov, starost ob pra-sitvi, dolžino predhodne laktacije ter povprecje in modus za poodstavitveni premor v nizih podatkovDS1 in DS2 po farmah

Farm Category No. of NBA AF (days) PLL (days) WCI (days)records x σ x σ x σ x mode

Gilts 17544 9.10 ± 2.79 350.0 ± 21.5 - -A Sows DS1 37134 10.58 ± 2.85 745.5 25.1 ± 6.7 11.0 5

Sows DS2 52490 10.49 ± 2.82 847.3 24.9 ± 6.4 11.1 5

Gilts 27243 8.84 ± 2.86 350.2 ± 33.5 - -B Sows DS1 71116 10.29 ± 2.87 767.1 26.6 ± 4.7 13.1 5

Sows DS2 90836 10.14 ± 2.90 902.6 26.5 ± 4.7 13.5 5

Gilts 26809 9.05 ± 2.47 333.8 ± 17.9 - -C Sows DS1 57913 10.17± 2.68 716.1 24.5 ± 5.1 12.2 5

Sows DS2 72602 10.13 ± 2.67 841.2 24.5 ± 5.1 11.7 5

The largest litter size was on farm A for both data sets (Table7). Average litter size in gilts ranged

among 8.84 piglets born alive on farm B and 9.10 on farm A. Variability of litter size was slightly

smaller in gilts than in sows. Standard deviation for NBA wassmaller on farm C compared to other

two farms. On farms A and B, age at first farrowing was similar (around 350 days), while on farm


Table 8: Number of records, mean and standard deviation for the number of piglets born alive (NBA),previous lactation length (PLL) and mean and mode for weaning to conception interval (WCI) by sowgenotype and farm for data set DS2Tabela 8: Število meritev, povprecje in standardni odklon za število živorojenih pujskov (NBA),dolžino predhodne laktacije (PLL) ter povprecje in modus za poodstavitveni premor (WCI) pogenotipu svinje po farmah v nizu podatkov DS2

Farm Sow genotype No. of NBA PLL (days) WCI (days)records x σ x σ x mode

SL 41696 10.07 ± 2.89 25.37 ± 6.92 11.35 5A SL x LWa 22652 10.41 ± 2.81 24.60 ± 6.02 10.66 5

LW x SLa 120 10.85 ± 2.95 24.85 ± 6.10 8.85 5LW 5566 9.50 ± 2.95 22.96 ± 3.84 11.55 5

SL 45960 9.54 ± 2.95 26.18 ± 4.65 14.67 5B SL x LWa 54679 10.19 ± 2.89 26.92 ± 4.86 12.81 5

LW x SLa 4567 10.19 ± 2.82 26.55 ± 4.33 12.04 5LW 12873 9.29 ± 2.96 26.28 ± 4.55 13.04 5

SL 43222 9.80 ± 2.68 24.21 ± 4.61 12.14 5C SL x LWa 52315 9.93 ± 2.65 24.80 ± 5.64 11.53 5

LW 3874 9.14 ± 2.62 23.57 ± 3.32 10.59 5

SL - Swedish Landrace; LW - Large White;aBreed of sow given first

C gilts farrowed approximately 16 days earlier. Variability of age at first farrowing differed among

farms, and it was smaller on farm C than on farms A and B. Weaning policies and weaning age

varied among farms too. On average, piglets were weaned between the 24th and the 27th days. Sows

conceived 11.0 to 13.5 days after weaning. Most frequently sows conceived on the fifth day after

weaning. Observed variables (NBA, PLL and WCI) did not differ substantially between smaller and

larger data sets within farm due to relatively small number of records added in data sets DS2.

Four sow genotypes were included: Swedish Landrace (SL), Large White (LW) and both crossbreeds

between them, except for farm C where genotype LW x SL was not presented in crossbreeding scheme

(Table 8). The largest number of sows were from SL and SL x LW genotype. Litter size was higher

for crossbred than for purebred sows. The lactation length did not differ between the genotypes

within farm, except for farm A. On farm A, in the last few yearsmanagement for LW sows included

shorter lactation length. The longest lactation was recorded on farm B. The weaning to conception

interval differed between farms, and it was the longest on farm B. Crossbred sows conceived earlier

than purebreds. Sows of all genotypes conceived most frequently on the fifth day after weaning.

Differences among farms can be the consequence of differentmanagement practices applied on farms.

For the illustration, on farm A, use of boars for oestrus detection was very successful, resulting in

larger litter size. On farm C, higher selection pressure on meat yield could be one of the reasons for

smaller litter size.

Litter size increased considerably up to the third parity, it reached the plateau between the third and

the fifth parity, and later decreased (Figure 4). The decrease of litter size after the fifth parity was


slower than increase up to the third parity. Litter size was larger along the whole trajectory, and

decreased slower for farm A than for the other two farms. Differences in the phenotypic variance for

the number of piglets born alive along trajectory by farms exist, but are small. In this sense, litter size

differs from growth and milk traits, where the averages and variances change much more along the

trajectory. The pattern of phenotypic variance changes andis specific for separate farm. Phenotypic

variance for the number of piglets born alive by parities wasthe lowest on farm C.

6

7

8

9

10

11

1 2 3 4 5 6 7 8 9 10

A

B

C

Num

ber

of

pig

lets

born

ali

ve

Parity

Variance

Average

Farm:

Figure 4: Averages and phenotypic variances for the number of piglets born alive by paritySlika 4: Povprecja in fenotipske variance za število živorojenih pujskov po zaporednih prasitvah

The pedigree file was prepared for three generations. Pedigree contained 24423 animals for farm A,

39405 animals for farm B, and 33389 animals for farm C (Table 9). All animals with records were

included in pedigree files. Pedigree files differed between farms in the number of ancestors, number

of base animals, and number of sires. The largest proportionof ancestors was in pedigree file for farm

B (28.01 % of animals), and the smallest for farm C where ancestors presented 16.40 % of pedigree

records. The proportion of base animals on farms A and C on oneside (from 3.35 to 3.52 % of total

number of animals) differed from farm B on the other side. Farm B had around 10 % of animal with

unknown parents from the total number of animals in pedigreefile. The number of sows per sire

differed among farms. On average there were between 25 sows per sire on farm A and 45 sows per

sire on farm C. Farms A and C had similar proportion of sires (22 and 23 %) that had only one sow,

while on farm B about 40 % of sires had only one sow.


Table 9: Pedigree structure within farmsTabela 9: Struktura porekla po farmah

FarmItem A B C

No. of animals with records 19170 28364 27912No. of ancestors 5253 11041 5477No. of base animals 820 3973 1176No. of sires 775 839 634No. of animals per sire (range) 1 - 372 1 - 520 1 - 1101

3.2 DATA FILE ORGANIZATION

Data were prepared differently for repeatability model andrandom regression model on the one side,

and for multiple-trait model on the other side. For repeatability model and random regression model,

records contained data for one litter only. Records for multiple-trait analysis contained information

for all litters related to one sow. Although effects were different between the first and higher parities,

specific data preparation (Table 10) allowed different models for the same trait in the repeatability

and random regression models. To illustrate this, previouslactation length was set to zero for gilts

to eliminate its effect, which did not exist, from the model.On the other hand, the second and later

parity sows have information about the previous lactation length. To combine records for gilts and

sows, previous lactation length in sows was adjusted with mean for the previous lactation length.

Therefore, gilts were put in the centre of regression, wherethey had no effect on the estimation of

regression coefficients. For weaning to conception interval effect, gilts were assigned to special class

"GILT". This class was dropped out from the analysis, since its equation was actually the same as the

equation for the first parity (gilts).

Table 10: Representation of prepared data structureTabela 10: Izsek iz pripravljenih podatkov

. Parity Age at farrowing Previous lactation length Weaningto conception interval NBA .

. . . . . . .

. 1 332 0 GILT 8 .

. 1 335 0 GILT 9 .

. 2 490 24-x 3 10 .

. 3 650 22-x 4 11 .

. . . . . . .


3.3 METHODS

Methods included procedures and rules applied to development of the model. Different possibilities

for fixed effects were presented. Implemented models were given in scalar, as well as matrix nota-

tion. The expected values and covariance structure by models were presented as matrix. Finally, the

description of eigenvalues for covariance matrices and breeding value prediction was described.

3.3.1 Model development

Statistical model was developed on larger data sets DS2 for three farms. The initial list of effects was

created on the basis of literature review. Alternatives studied were described below. For some effects,

several alternatives were tried. Some effects were very simply modelled, while modelling of other

effects changed during the study and required a more time. The following possibilities for effects

were checked:

Sow genotypewas applied as class effect with four levels, two for purebreds and two for crossbreeds

between them.

Sire genotypehad between six and eight levels by farm denoting purebred orcrossbred service sires.

Seasonwas defined as year-month interaction for mating season.

Service sirewas implemented by two different ways. At first, it was treated as random effect. While

low proportion of phenotypic variance was explained, we implemented the model with service sire as

fixed effect. Service sires with less than 10 litters were grouped within genotype.

Weaning to conception interval was modelled using few different ways. Linear regression was

simple but did not describe changes in litter size sufficiently for litters conceived between day 6 and

10. The drop in litter size five days after weaning was modelled with different functions. Due to

irregularity in changes in litter size with prolongation ofweaning to conception interval, weaning to

conception interval was split into 10 classes. Therefore, WCI was modelled as:

- linear regression

- different functions: polynomials of different order, modified lactation curve

- class effect with ten levels: 1 - 3, 4, 5, 6, 7, 8, 9, 10 - 23, 24 -33, and 34 - 70 days.

Previous lactation lengthwas further used as linear regression by two ways. Firstly, as a simple

linear regression along the whole interval, and secondly asa linear regression nested within three

intervals. Intervals of previous lactation length were defined as a consequence that linear regression

fitted good relationship between lactation length and litter size only in the middle of interval. Outside

this interval linear regression was not suitable enough. Asa result the previous lactation length was

modelled as:


- linear regression along the whole interval

- linear regression nested within three intervals (1 - 18, 19- 34, 35 - 60 days)

Parity was modelled as:

- class effect with ten levels,

- class effect with three levels: for the first (1), second (2), and higher parities (3+).

Age at farrowing was fitted as:

- linear regression, only for the first parity and for all parities

- quadratic regression, only for the first parity and for all parities

- quadratic regression nested within three parity classes (1, 2, 3+).

Due to simplicity and descriptive overview the most important models were presented in Table 11.

Model selection was based on significance of the effects, coefficient of determination and degrees

of freedom for the model. The proportion of variation explained by effect was studied, as well as

simplicity and interpretation of the model. From the all effects that existed in the data, only effects

significant on 0.05 level were included in the model. The criteria was high because of the size of

the data. Additionally, the choice of effects depended on fact that some effects in the data were

autocorrelated. Effects which were not significant, as wellas effects which partly presented other

effect were sequentially dropped from the model.

Table 11: Modelling of the fixed effectsTabela 11: Modeliranje sistematskih vplivov

Model EffectsSow genotype Mating season Service sire WCI PLL AF Parity

A CL CL - LR LR LR CLB CL CL CL LR LR LR CLC CL CL CL LR LR LR* CLD CL CL CL LR LR QR* CLE CL CL CL CL LR QR CLF CL CL CL CL LR QR CL**

Final CL CL CL CL LR QR(CL**)

QR - quadratic regression; LR - linear regression; CL - classeffect; AF - age at farrowing; WCI - weaning to conception

interval; PLL - previous lactation length; *only for first parity; **three classes

Fixed part of the model was developed using least square method as implemented in GLM procedure

from the statistical package SAS/STAT (SAS Inst. Inc., 2001). Excluding one by one effect form the

final model, a list of effects by deviation fromR2 of the final model was created. Larger deviation

in R2 for individual effect from theR2 obtained with full model means that this effect had a larger

influence on litter size. Deviation from theR2 was presented in percentage units.


3.3.2 Implemented models

Three types of the model were implemented: repeatability model, multiple-trait model, and random

regression model. Repeatability model was developed because it is a frequent procedure used for

genetic evaluation in litter size. Multiple-trait model was usually used as theoretic alternative to

repeatability model, while random regression model represented new approach in dealing with longi-

tudinal data.

a) Repeatability model

Repeatability model can be applied whenever we can assume the complete genetic correlation be-

tween litter size in different parities. It was also performed for the first validation of the effects

included in the model as well as data structure. Repeatability model was applied for data set DS2.

The following repeatability model (2) in scalar notation was used:

yijklmn = µ + Pi + Gj + Sk + Bl + Wm + bI i (xijklmn − x) + bII i (xijklmn − x)2 +

+ bIII (zijklmn − z) + ljl + pjn + ajln + eijklmn

... (2)

where

yijklmn - number of piglets born alive,

Pi - parity class,

Gj - sow genotype,

Sk - mating season,

Bl - service sire,

Wm - weaning to conception interval,

bI m, bII m- linear and quadratic regression coefficients for age at farrowing nested within parity class,

xijklmn - age at farrowing,

bIII - linear regression coefficient for previous lactation length,

zijklmn - previous lactation length,

ljl - common litter environmental effect,

pjn - permanent environmental effect,

ajln - direct additive genetic effect,

eijklmn - residual term.

The random part of the repeatability model consisted of a common litter environmental effect(l),

permanent environmental effect(p), and direct additive genetic effect(a). Different random effects


were tried in repeatability model. Its inclusion in the random part of the model was based on the size

of their variance components. Development of the random part of the model showed a negligible

estimate for the maternal genetic effect and it was not included in the model. Therefore, in the

model without maternal effects, common litter environmental effect includes beside environmental

components genetic effects too.

b) Multiple-trait model

The multiple-trait model considers litter size in successive parities as different traits and allows the

evaluation of covariance components for random effects between parities. Multiple-trait model was

performed in order to get the estimate of covariance structure between litter size in different parities

for the comparison with covariance structure obtained later with random regression model. The anal-

ysis was conducted only for the first six parities i.e. data sets DS1 due to computational problems in

multiple-trait analysis for data sets DS2. The appropriatemultiple-trait models in the scalar notation

were as follows:

yijklm = µ + Gij + Sik + Bil + bI i (xijklm − x) + bII i (xijklm − x)2 +

+lijl + ailm + eijklm

... (3)

yijklmn = µ + Gij + Sik + Bil + Wim + bI i (xijklmn − x) + bII i (xijklmn − x)2 +

+bIII i (zijklmn − z) + lijl + ailm + eijklmn

... (4)

where

yijklm(n) - number of piglets born alive,

i - index for parity,i = 1 in model (3) for gilts, andi = 2 - 6 in models (4) for sows,

Gj - sow genotype,

Sk - mating season,

Bl - service sire,

Wm - weaning to conception interval,

bI , bII - linear and quadratic regression coefficients for age at farrowing,

xijklmn - age at farrowing,

bIII - linear regression coefficient for previous lactation length,

zijklmn - previous lactation length,

lijl - common litter environmental effect,

ailm - direct additive genetic effect,

eijklmn - residual term.


Number of piglets born alive at each parity was treated as a different trait, and thus resulting in six

traits. The fixed part of the model for the first parity (3) differed from the one for higher parities

(4). Only the model for sows contained the previous lactation length and weaning to conception in-

terval. In multiple-trait analysis, permanent environmental effect cannot be separated from residual.

Therefore, the random part of multiple-trait model consisted of direct additive genetic effect(a)and

a common litter environmental effect(l).

c) Random regression model

Covariance functions have been proposed as an alternative solution to deal with litter size over parities

as a longitudinal trait. Orthogonal polynomials representthe coefficients most widely used to describe

a trait as a function of age and they have been chosen as suitable functions to represent coefficients for

the covariance functions and for random regression models.Orthogonal polynomials of standardized

units of time have been recommended as covariables. The primary general advantage is the reduced

correlation among the estimated coefficients. Several types of orthogonal polynomials are available,

but Legendre polynomials were usually used. Legendre polynomials do not require prior assumptions

about the shape of trajectory and are the easiest to calculate.

The following random regression model (5) with Legendre polynomials of standardised parity was

fitted:

yijklmn = µ + Pi + Gj + Sk + Bl + Wm + bI i (xijklmn − x) + bII i (xijklmn − x)2 +

+bIII (zijklmn − z) +∑3

s=1

∑vt=0 αs tφt(p

∗ijklmn) + εijklmn

... (5)

The time variable in random regression model for litter sizeis parity. Orthogonal Legendre polyno-

mials from linear (LG1) with two terms to cubic (LG3) with four terms were fitted. The other term

used for power of LG is the order of polynomial and it is equal to the number of terms in polynomial.

The set of fixed effects used in the RRM analyses for NBA included the same effects as for above

mentioned repeatability model. The direct additive genetic effect (s=1), permanent environmental ef-

fect (s=2), and common litter environmental effect (s=3) were fitted as random regressions on parity

using corresponding Legendre polynomials (φt(p∗ijklmn)). The standardized parity (p∗ijklmn), with

range from -1 to +1, was derived from (6) wherepmin is the first andpmax the last (sixth or tenth)

parity.

p∗ijklmn =2 (p − pmin)

(pmax − pmin)− 1 ... (6)

3.3.3 Models in matrix notation and covariance structure

The three models with their expected values and (co)variance structure were presented in matrix

notation.


a) Repeatability model

The repeatability model applied in our investigation can bewritten in matrix notation as shown in

Eq. 7:

y = Xβ + Zll + Zpp + Zaa + e ... (7)

wherey is vector of observations,X is incidence matrix for fixed effects,β is vector of unknown

parameters for fixed effects,Zl is incidence matrix for common litter environmental effect, l is

vector of parameters for common litter environmental effect, Zp is incidence matrix for permanent

environmental effect,p is vector of parameters for permanent environmental effect, Za is incidence

matrix for direct additive genetic effect,a is vector of parameters for direct additive genetic effect,

e is vector of residuals.

The following expected values (Eq. 8) and structure of covariance matrices (Eq. 9 and Eq. 10) were

assumed in repeatability model:

E

y

l

p

a

e

=

Xβ

0

0

0

0

... (8)

var (y) = ZlGlZ′

l + ZpGpZ′

p + ZaGaZ′

a + R ... (9)

var

l

p

a

e

=

Ilσ2l 0 0 0

0 Ipσ2p 0 0

0 0 Aσ2a 0

0 0 0 Ieσ2e

Gl 0 0 0

0 Gp 0 0

0 0 Ga 0

0 0 0 R

... (10)

whereA is the numerator relationship matrix,Il is identity matrix for common litter environmental

effect, Ip is identity matrix for permanent environmental effect,Ie is identity matrix for residual.

MatricesIl, Ip, andIe declare that common litter environmental effect, permanent environmental

effect, and residual are identically and independently distributed. Covariances between random

effects were assumed to be zero.

b) Multiple-trait model

The multiple-trait model in matrix notation contains another list of unknown parameters (βM ) ac-

companied by incidence matrix (XM ) as presented in Eq. 3 and Eq. 4.

y = XMβM + Zll + Zaa + e ... (11)


Structure of expected values (Eq. 12) and covariances (Eq. 14) in multiple-trait model were as follows:

E

y

l

a

e

=

XMβM

0

0

0

... (12)

var (y) = ZlGlZ′

l + ZaGaZ′

a + R ... (13)

var

l

a

e

=

Il⊗Gl 0 0

0 A⊗ Ga 0

0 0∑⊕

R0i

... (14)

In multiple-trait model, litter size in each parity was assumed to be a different trait that resulted in six

trait models. Covariance components for direct additive genetic effect were obtained as direct product

between numerator relationship matrix (A) and matrix of additive genetic covariances (Ga) among

traits measured on the same animal. Structure of covariancematrix for common litter environmental

effect (Gl) is similar as for Ga (Eq. 15). Residuals among animals were independent and within

traits identically normally distributed. Residual variance (R) is a direct sum of residual matrices (R0i)

for a separate trait (Eq. 16). Traits measured on different animals are independent while covariances

among residuals for measurements on the same animal do exist. The number of litters per animal

differs, mainly due to culling, resulting in variance matrix for each missing pattern.

Ga =

σ2a1 σa12 σa13 σa14 σa15 σa16

σ2a2 σa23 σa24 σa25 σa26

σ2a3 σa34 σa35 σa36

σ2a4 σa45 σa46

Sym. σ2a5 σa56

σ2a6

... (15)

R0i =

σ2ei 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

i = 1, ..6 ... (16)

c) Random regression model

The random regression model in matrix notation (Eq. 17) seems similar to repeatability model in

Eq. 7. However, the list of unknown parameter for random effects differs from comparison of Eq. 2

and Eq. 5 as follow:

y = Xβ + Zll + Zpp + Zaa + ε ... (17)


wherey is vector of observations,X is incidence matrix for fixed effects,β is vector of unknown

parameters for fixed effects,Zp is incidence matrix for permanent environmental effect,p is vector

of parameters for permanent environmental effect,Zl is incidence matrix for common litter environ-

mental effect,l is vector of parameters for common litter environmental effect,Za is incidence matrix

for direct additive genetic effect,a is vector of parameters for direct additive genetic effect,ε is vector

of residuals.

Following structure of expected values (Eq. 18) and covariances (Eq. 19) were assumed:

E

y

l

p

a

ε

=

Xβ

0

0

0

0

... (18)

var (y) = ZlKlZ′

l + ZpKpZ′

p + ZaKaZ′

a + Rε

var

l

p

a

ε

=

Il ⊗ K0l 0 0 0

0 Ip ⊗ K0p 0 0

0 0 A⊗ K0a 0

0 0 0∑⊕

R0i

... (19)

whereA is the numerator relationship matrix,K0a is the covariance matrix for direct additive ge-

netic effect (Eq. 20),K0l is the covariance matrix for common litter effect,K0p is the covariance

matrix for permanent environmental effect,Il andIp are identity matrix, andR0i is residual matrix.

Symbol⊗ represents Kronecker (direct) product, and symbol∑⊕ denotes direct sum. Structure of

covariance matrix for common litter environmental effect (K0l) and covariance matrix for permanent

environmental effect (K0p) is similar as forK0a.

K0a = var

αt 0

αt 1

...

αt tA−1

=

σ2α0 σα01 · · · σα0(tA−1)

σ2α1 · · · σα1(tA−1)

. . ....

σ2αtA−1

... (20)

Covariances for litter size records for every combination of two parities were calculated as shown in

the case of direct additive genetic effect (21).

Ca = ΦKaΦ′

... (21)

whereCa is a matrix of covariances andΦ is a matrix containing covariables for Legendre polyno-

mials for parities.

Estimation of the covariance components in multiple-traitand random regression models was based

on Restricted Maximum Likelihood Method (REML) using the VCE-5 software package (Kovac


et al., 2002). The analytical gradients of the likelihood function are explicitly calculated in op-

timization procedure for maximizing the likelihood. In theestimation of genetic parameters, the

convergence criterion was set to10−6 for all analyses. Modules SAS/IML and SAS/GRAPH from

the statistical software package SAS (SAS Inst. Inc., 2001)were used for the visualization and 3D

graphical analysis.

3.3.4 Eigenvalues and eigenfunctions

Covariance functions can be used to analyze patterns of inheritance in the covariance matrices (Kirk-

patrick and Heckman, 1989; Kirkpatrick et al., 1990). For this purpose, eigenvalues and eigenfunc-

tions were determined. Module SAS/IML (SAS Inst. Inc., 2001) was used for computation of eigen-

values for covariance matrices of regression coefficients obtained in random regression models with

different order of Legendre polynomials. Eigenvalues and corresponding eigenvectors were calcu-

lated from covariance matrices. Eigenvalues of genetic andenvironmental covariance matrices quan-

tify the relative importance of each order of Legendre polynomials. Eigenvalues for direct additive

genetic effect, permanent environmental effect and commonlitter environmental effect were calcu-

lated and presented in absolute value and in relative value as percentage.

To obtain the eigenfunction coefficients, the matrix of the first four Legendre polynomials was mul-

tiplied by eigenvector obtained from covariance matrix of random regression coefficients with cubic

power. As a result, matrix four by four was obtained, where the first column presents eigenfunc-

tion coefficients for the first eigenfunction, the second column presents coefficients for the second

eigenfunction etc.

3.3.5 Breeding values

Random regression coefficients for direct additive geneticeffect could be used for the prediction of

breeding value at any point along the trajectory for each animal. Random regression model with dif-

ferent power was used to decide which order of Legendre polynomials is sufficient for the analysis

and for obtaining regression coefficients for each animal. Random regression coefficients are param-

eters of production function and they determine the course of litter size during lifetime of the sow

(Eq. 22).

ap =3

∑

s=1

v∑

t=0

αs tφt(p∗ijklmn) ... (22)


4 RESULTS

The results of the study will be presented in two main parts. The first part of the results presents

the choice of the model and estimates of fixed effects obtained with the data sets DS2 for the three

farms. The second part of the results includes estimates of genetic parameters using repeatability,

multiple-trait and random regression models. In RRM approach beside the estimation of covariance

components, special attention was placed on eigenvalue analysis and utilisation of random regression

coefficient in the prediction of breeding values for litter size.

4.1 MODEL SELECTION

The selection of statistical model included the choice of fixed effects followed by the choice of random

effects. Determination of the effects included in the modelwas conducted by repeatability model.

Later, effects selected in repeatability model were also used in multiple-trait and random regression

model. The choice of the model was limited by effects collected by litter recording scheme.

4.1.1 Fixed effects

Alternative fixed effects were fitted in the model for the number of piglets born alive. There were

many combinations, but only seven models (A - G) were presented (Table 11).The models usually

explained less than 10 % of total variation. The decisions were based mainly on the coefficient of

determination (R2) and the number of degrees of freedom (d.f.) for the model (Table 12).

The simplest model (A) included sow genotype, parity, and mating season as class effects. Weaning

to conception interval, previous lactation length and age at farrowing were considered as linear re-

gression. The obtainedR2 was the smallest in model A in relation to other models for allthree farms.

TheR2 increased considerably for all farms by adding service sirein model B. Although the d.f. for

the model increased from 535 on farm A to 621 on farm B, the effect showed considerable deviations

among service sires. The service sire explained between 1.2% of variability on farm C and 1.7 % on

farm A. The models C, D, E, and F varied in weaning to conception interval, age at farrowing and par-

ity (Table 11, page 32). Weaning to conception interval was fitted as linear regression in models C and

D, and as class effect in models E and F. Age at first farrowing was presented as linear regression in

model C and as quadratic regression in model D. In models E andF, age at farrowing was considered

as quadratic regression. TheR2 was slightly enlarged by model complexity. The smallest increase

of R2 was observed with changing of linear with quadratic regression for age at first farrowing from

model C to D. The improvement from model B to the final model (G)was moderate, and ranged from

0.7 % on farm C to 1.0 % on farm B for six additional parameters to be estimated. The final model (G)

which included sow genotype, mating season, service sire and weaning to conception interval as class

effects, previous lactation length as linear regression, and age at farrowing as quadratic regression

nested within parity class, had the highestR2 for all three farms.


Table 12: Coefficients of determination (R2) and degrees of freedom (d.f.) for different models perfarmTabela 12: Koeficienti determinacije (R2) in stopnje prostosti (d.f.) za razlicne modele po farmah

Model Farm A Farm B Farm C

R2 d.f. R2 d.f. R2 d.f.

A 0.078 147 0.076 171 0.076 174

B 0.095 682 0.094 792 0.088 719

C 0.098 682 0.099 792 0.090 719

D 0.098 683 0.100 793 0.091 720

E 0.101 691 0.100 801 0.093 728

F 0.100 684 0.098 794 0.093 721

G 0.103 688 0.104 798 0.095 725

Between 9.5 and 10.4 % of total variability was explained forthe number of piglets born alive with

fixed part in model G on the three farms (Table 12). The number of degrees of freedom for the model

ranged between 688 and 798 (Table 13). Larger number was mainly a consequence of two effects:

mating season and service sire. Mating season defined as yearmonth interaction had 133 levels on

farm A, 157 levels on farm B, and 161 levels on farm C. The number of levels for service sire effect

ranged between 536 on farm A and 622 on farm B. Root mean squareerror (σe) was similar to that

between farms A and B, and smaller for farm C (Table 13).

Table 13: Number of levels for mating season and service sireeffect, degrees of freedom (d.f.) formodel, coefficient of determination (R2) and standard deviation (σ) on three farmsTabela 13: Število nivojev za sezono pripusta in merjasca oceta gnezda, stopnje prostosti za model(d.f.), koeficient determinacije (R2) in standardni odklon (σ) po farmah

Farm Number of seasons Number of sires d.f. R2 σ

A 133 536 688 0.103 2.74B 157 622 798 0.104 2.79C 161 546 725 0.095 2.54

In the final model, all the included effects were significant (P<0.001). In addition, a change in the

coefficient of determination (△R2) from the R2 obtained with full final model that included all

chosen effects was calculated (Table 14).

Two effects with the largest change from theR2, when individual effect was excluded from the final

model, were the age at farrowing nested within parity class and service sire. The change inR2 ob-

tained with the full model was the largest for the age at farrowing nested within parity class effect on

two farms (B and C) and ranged between 2.10 and 2.30 percentage units. Excluding the service sire

effect from the full model resulted in a decrease of theR2 between 1.20 and 1.73 percentage units.

Considerably smaller decrease ofR2 was obtained with the elimination of weaning to conception


Table 14: Change in the coefficient of determination (∆R2) from theR2 obtained with full model forindividual effect per farmTabela 14: Sprememba koeficienta determinacije (∆R2) od koeficienta determinacije polnega modelaza posamezne vplive po farmah

FarmEffect A B CAge at farrowing nested within parity 0.88 2.30 2.10Service sire 1.70 1.73 1.20Weaning to conception interval 0.50 0.57 0.57Mating season 0.52 0.40 0.40Previous lactation length 0.40 0.31 0.17Sow genotype 0.26 0.18 0.23

interval and mating season from the model. Change inR2 obtained with full model for these two

effects ranged between 0.40 and 0.57. Two effects of which exclusion from the full model resulted in

the smallest change inR2 were previous lactation length and sow genotype. Excludingthese effects

separately from the full model that included all fixed effects resulted in decrease ofR2 between 0.17

and 0.40 percentage units.

Intensive analysis using repeatability model showed that the following fixed effects could be used in

statistical model for genetic evaluation of litter size on all three farms. Fixed effects included in the

model for the number of piglets born alive were: sow genotype, mating season, service sire, weaning

to conception interval, parity class, age at farrowing, andprevious lactation length. The sow genotype,

mating season, service sire, weaning to conception interval and parity were fitted as class effects. The

mating season was defined as year month interaction. The weaning to conception interval was defined

as class effect with the following classes: 1 - 3, 4, 5, 6, 7, 8,9, 10 - 23, 24 - 33, and 34 - 70 days.

Three classes were considered for parity effect: the first for parity one, the second for parity two, and

the third for later parities. Age at farrowing was modelled as quadratic regression and the previous

lactation length was fitted as linear regression.

Estimates of fixed effects obtained with the final model will be presented in the form of tables and

figures for effects with a relatively small number of levels and effects fitted as regressions. Effects

with a large number of levels were presented only graphically. The order of effects was presented in

accordance with deviation fromR2 of the full model (Table 14).

4.1.2 Age at farrowing and parity

Age at farrowing can be expressed chronologically or as parity. The presentation of age in parities

is simple, but is not always sufficient. The number of pigletsborn alive increased by parity to

a specific age that coincide with the parity three to five. However, age at farrowing within each

parity varied considerably. The range increased by parity.Wide range of possible ages at farrowing


within parity was a reason to combine these two effects. Graphical presentation of these two ef-

fects (Figure 5) clearly showed some difference in pattern between the first two and higher parities.

Litter size increased clearly with age at farrowing in the first two parities. The increase should be

presented by quadratic regression. Litter size within parity increased faster, when gilts or sows were

younger, and slower when they were older. After the second parity, litter size within parity was

decreasing and could be presented very well with the same curve. Some exception was noticed only

for the last parity curves, probably due to small number of data in the last parities and characteris-

tics of quadratic regression. Apparently, the common function could describe all ages after parity two.

7.0

7.5

8.0

8.5

9.0

9.5

10.0

10.5

11.0

200 400 600 800 1000 1200 1400 1600 1800 2000

Nu

mb

er o

f p

igle

ts b

orn

ali

ve

Age at farrowng (days)

12

34

5

6

7

8

9

10

Figure 5: Relationship between age at farrowing and the number of piglets born alive nested withinparity for farm BSlika 5: Povezava med starostjo ob prasitvi in številom živorojenih pujskov znotraj zaporedne prasitvena farmi B

In the final model, age at farrowing was nested within parity where parities from three to ten were

combined into one class (Figure 6). Therefore, the effect ofage was sufficiently described by three

quadratic regressions describing changes in litter size within the first and second parity separately and

within the higher parities together. Litter size in the firstparity was increasing up to one year. The

increase was larger when gilts were younger and diminished as they become older. Differences in

the level and curve shape among farms were due to different mating policies and puberty stimulation.

Thereafter, litter size reached constant level or even decreased. Changes in the number of liveborn

piglets are similar, but slightly smaller and moved to the right as the sows are 150 days older. Thus,

litter size at the second farrowing increased up to the age ofaround 500 day.


7.0

7.5

8.0

8.5

9.0

9.5

10.0

10.5

11.0

200 400 600 800 1000 1200 1400 1600 1800 2000

1 2 3

Num

ber

of

pig

lets

born

ali

ve

Age at farrowing (days)

Parity class:

Farm A

7.0

7.5

8.0

8.5

9.0

9.5

10.0

10.5

11.0

200 400 600 800 1000 1200 1400 1600 1800 2000

1 2 3

Num

ber

of

pig

lets

born

ali

ve


Parity class:

Farm B

7.0

7.5

8.0

8.5

9.0

9.5

10.0

10.5

11.0

200 400 600 800 1000 1200 1400 1600 1800 2000

1 2 3

Num

ber

of

pig

lets

born

ali

ve


Parity class:

Farm C

Figure 6: Relationship between the age at farrowing and the number of piglets born alive nestedwithin three parity class per farmSlika 6: Povezava med starostjo ob prasitvi inštevilom živorojenih pujskov znotraj razredov za-poredne prasitve po farmah


After the second parity effect of age at farrowing substantially decreased, only a small change in

litter size within each parity was noticed. A substantial increase of litter size in the first parity is a

consequence of higher ovulation rate at a higher oestrus number. Litter size of sows that farrowed at

lower age at first farrowing was also smaller in the second parity, mainly due to unfinished maturation

process, as well as possible reproductive problems in gilts. After the second parity, the effect of

age on litter size reduced, although a curve for the third parity shows similar pattern as for the first

two parities. The plateau of litter size was reached betweenthe third and the fifth parity. Litter size

decreased after the plateau because of ageing of sows and consequently lower ovulation rate. In the

last few parities, litter size can increase again due to lower number of sows in the last parities.

Litter size was decreasing after the fourth parity. Age at farrowing within parity showed a decreasing

trend as well. There is no evidence that in later parities both effects are needed. Simplified model

where age at farrowing was the only effect used for the third and later parities proved to be a good

solution. Shape of the curves for litter size by age at farrowing nested within three parity classes was

similar between farms. Litter size was slightly larger for farm A than for other two farms (Figure 6)

and the curve for litter size for the third parity class declined slower than litter size curves for farms B

and C. Litter size curves for the first parity class differed in width, while farm C curve shifted a little

to the left. This could be connected with the fact that sows onfarm C mated earlier and therefore they

farrowed younger on the average.

Table 15: Estimates of regression coefficients with standard errors (SEE) for age at farrowing nestedwithin parity class per farmTabela 15: Ocene regresijskih koeficientov in njihove standardne napake (SEE) za starost ob prasitvipo zaporednih prasitvah in farmah

Farm Parity class Linear coefficient Quadratic coefficient

Estimate + SEE p value Estimate + SEE p value

1 0.1485± 0.0238 <0.0001 -0.00019±0.000033 <0.0001

A 2 0.0914± 0.0246 0.0002 -0.00008±0.000024 0.0005

3+ 0.0032± 0.0004 <0.0001 -0.00001±0.000001 <0.0001

1 0.1279± 0.0091 <0.0001 -0.00016±0.000012 <0.0001

B 2 0.0501± 0.0107 <0.0001 -0.00004±0.000010 <0.0001

3+ 0.0025± 0.0002 <0.0001 -0.00001±0.000001 <0.0001

1 0.1987± 0.0215 <0.0001 -0.00027±0.000031 <0.0001

C 2 0.0333± 0.0222 0.1339 -0.00002±0.000022 0.2399

3+ 0.0023± 0.0030 <0.0001 -0.00001±0.000001 <0.0001


The regression coefficients (linear and quadratic) were larger for the first parity class compared to

higher parity classes (Table 15), while coefficients by parity class were similar among farms (Ta-

ble 15). Estimates of linear regression coefficients were positive. Linear regression coefficients for

age at farrowing nested within parity class ranged between 0.13 and 0.20 for parity class one, between

0.03 and 0.09 for parity class two, and between 0.002 and 0.003 for the third parity class that included

later parities. Some higher values of estimate for the first parity on farm C could be explained as a

consequence of lower mating age applied. Namely, substantial increase in litter size in gilts presented

with the slope of quadratic curve may be a result of mating mainly at the first and the second oestrus.

Estimated quadratic regression coefficients were negativeand smaller compared to linear ones, as

expected. Quadratic regression coefficients showed similar tendency of decrease by parity class as

linear regression coefficients.

4.1.3 Service sire

The effect of service sire presents a direct effect on the number of piglets born alive. It differed from

the effect of sire from pedigree where it describes indirecteffect on his daughter’s litter size. Service

sire modelled as fixed or trivial random effect included genetic, as well as environmental (disease,

mating frequency, welfare,...) components.

-3.00

-2.00

-1.00

0.00

1.00

2.00

Nu

mb

er o

f p

igle

ts b

orn

ali

ve

Service sire

Figure 7: Estimates of the service sire effect on the number of piglets born alive for farm BSlika 7: Ocene za vpliv merjasca - oceta gnezda za število živorojenih pujskov na farmi B

Estimates for farm B among service sires for the number of piglets born alive showed a considerable

variation (Figure 7). Similar variation in NBA was found in the other two farms. The difference

between the best and the worst estimate was even 6.3 livebornpiglets on farm A, 5.3 liveborn piglets


on farm B, and 4.9 liveborn piglets on farm C. The sires had at least 10 litters, otherwise they were

grouped into one class for each genotype. The service sires with the best estimate had on average 40

litters, and sires with the worst estimates even more than 100 litters. The differences were consider-

able and can not be declared as random drift. The majority of the estimates for service sires (more

than 90 %) was observed inside interval from -1 to +1 piglets born alive for all farms. Due to relatively

large number of litters per service sire, estimates for themcould be considered as reliable. Therefore,

a possibility exists to maintain litter size at a satisfactory level by culling the less prolific service sires.

More attentions should be paid on service sires with extremenegative and positive values.

4.1.4 Weaning to conception interval

Weaning to conception interval distribution varied among farms. Weaning to conception interval

showed three peaks on all farms observed (Figure 8). The firstthree days after weaning a small

proportion of sows was sired (less than 1 %). Very often oestrus appears on day 4, where sows

conceived in 6 % cases on farm A to 14 % cases on farm B. The first and the highest peak happened

on day 5 with between 40 % on farm B and 54 % on farm A of successfully mated sows. The

percentage of sows conceived on day 5 was the highest on farm A, and amounted to more than 50 %

of sows. Between 54 and 60 % sows were successfully mated within the first 5 days after weaning.

The percentage of services on day 6 dropped to less than 12 %. The decrease continued up to day 10.

However, between 16 and 18 % of sows were mated from the day 6 to10 after weaning. Altogether,

three quarters of sows become pregnant before day 10. The next peak could be observed 21 days

later, appeared at around 25 days after weaning and lasting about 5 days. At the second peak, only

2 % of sows conceived per day. There were less successful matings between the two peaks. The last

peak was hardly observed because of the culling policy. If the sows are not pregnant after the second

regular oestrus, they are more often culled than mated again.

Evident pattern of NBA decrease was observed between the sixth and the tenth day of weaning to

conception interval (Figure 8) for all three farms. According to considerably smaller litters obtained

when sows enter in conception between the sixth and the tenthday after weaning, decreasing of

proportion of litters on this interval could increase overall litter size on farms. In relation to the fifth

day of weaning to conception interval, litter size had reduced by -0.68 liveborn piglets on farm A,

-0.42 liveborn piglets on farm B, and -0.75 liveborn pigletson farm C (Table 16). Litter size was

higher before day five than on the fifth day of the weaning to conception interval. On the sixth day of

WCI, litter size decreased. The smallest values for NBA wereobserved on the day seven to eight of

weaning to conception interval. After the day nine of WCI, litter size again increased and achieved

higher litter size level than before the day 5. A drop in litter size from the sixth to ninth day is

important from economical point of view, because it can include between 10 and 20 % of all litters.

The average number of piglets born alive showed a distinct decrease in the second five days after

weaning (Table 16). After the tenth day of WCI, litter size reached the previous level as before the

decrease.


9.0

9.5

10.0

10.5

11.0

11.5

0.0

10

20

30

40

50

60

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70

Num

ber

of

pig

lets

born

ali

ve

Freq

uen

cy (%

)

Weaning to conception interval (days)

Farm A

9.0

9.5

10.0

10.5

11.0

11.5

0.0

10

20

30

40

50

60

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70

Num

ber

of

pig

lets

born

ali

ve

Freq

uen

cy (%

)


Farm B

9.0

9.5

10.0

10.5

11.0

11.5

0.0

10

20

30

40

50

60

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70

Num

ber

of

pig

lets

born

ali

ve

Freq

uen

cy (%

)


Farm C

Figure 8: The average number of piglets born alive and distribution of weaning to conception intervalper farmSlika 8: Povprecno število živorojenih pujskov in porazdelitev poodstavitvenega premora po farmah

Lukovic

Z.C

ovariancefunctions

forlitter

sizein

pigsusing

arandom

regressionm

odel.D


Ljubljana,U

niv.ofL

jubljana,Bio

technicalFaculty,Z

ootechnicalDepartm

ent,200649

Table 16: Effect of weaning to conception interval (WCI) on litter size expressed as a deviation from day 5 per farmTabela 16: Vpliv poodstavitvenega premora (WCI) na velikost gnezda kot odstopanje od petega dneva po farmah

Farm A Farm B Farm C

WCI (days) Estimate + SEE p-value Estimate + SEE p-value Estimate + SEE p-value

1-3 0.065±0.151 0.6684 -0.105±0.131 0.4223 0.113±0.269 0.6742

4 0.119±0.055 0.0315 0.113±0.032 0.0005 0.109±0.032 0.0006

5 0.000 0.000 0.000

6 -0.427±0.037 <0.0001 -0.251±0.034 <0.0001 -0.214±0.034 <0.0001

7 -0.579±0.075 <0.0001 -0.428±0.062 <0.0001 -0.546±0.075 <0.0001

8 -0.685±0.132 <0.0001 -0.385±0.092 <0.0001 -0.759±0.075 <0.0001

9 -0.608±0.165 0.0002 -0.351±0.119 0.0034 -0.421±0.098 <0.0001

10-23 0.228±0.051 <0.0001 0.340±0.041 <0.0001 0.147±0.046 0.0017

24-33 0.291±0.044 <0.0001 0.496±0.039 <0.0001 0.429±0.034 <0.0001

>34 -0.006±0.051 0.1871 0.361±0.037 <0.0001 0.271±0.038 <0.0001

*SEE - standard error of the estimate


This level of litter size was maintained up to the fifty day, afterwards it oscillated more, probably due

to the smaller number of litters.

Weaning to conception interval is a continuous variable. Recording to some evidence litter size was

better when oestrus was expected and dropped substantiallyotherwise. Several different functions

were used to describe the association between WCI and littersize. The simplest, linear function did

not describe the drop of litter size on day 6 after weaning. Itwas not suitable to describe the connec-

tion between weaning to conception interval and litter size, especially in the first part of the interval.

Another approach included the use of polynomials of the second and the third power nested within a

few defined parts of the whole interval. The results were not promising. At last, the idea was found

in modeling the lactation curve (Guo and Swalve, 1995) usingseveral coefficients for describing the

relationship between litter size and weaning to conceptioninterval. The function (Figure 9) was com-

posed from several different transformations of independent variable like square root, cosines, cubic,

and exponential function. The function might be better compared to regression due to the allowed

specific change of litter size. However, any function which would fit the changes in litter size suffi-

ciently in various situations, was not found. Reduction on the sixth day, as well as the normalization

after day 10 were stepwise. After these, more or less unsuccessful attempts, weaning to conception

interval was treated as class effect.

9.5

10.0

10.5

11.0

11.5

0 10 20 30 40 50 60 70

Function

Means

Num

ber

of

pig

lets

born

ali

ve


Figure 9: Parametrical function for description of relationship between weaning to conception intervaland the number of piglets born aliveSlika 9: Parametricna funkcija za opis povezave med poodstavitvenim premoromin številom živoro-jenih pujskov

4.1.5 Mating season

Mating season was fitted as year-month interaction (Figure 10). There were evident long and short

term seasonal effects. Long term trends showed three patterns. On farm A, litter size decreased at

first, after that it achieved a certain level, and at the end itincreased. A decrease in litter size was


connected to undesirable changes in age structure and genetic structure. Considerable increase of litter

size on farm A in the last period was a consequence of mating management (oestrus synchronization

with boar, high farrowing rate). Litter size on farm B was on the same level for the whole period,

except for the last two years when litter size decreased. Along the whole period, litter size on farm

C increased. Continual increase in litter size is a consequence of hard persistent work. Although, the

level of litter size is similar on all three farms in mid period, long term seasonal trends for the number

of piglets born alive differed. Beside the usual short term oscillation, there was the evidence of slow

increase in litter size by farms from 1994 to 1999.

In the first few years, as well as in the last few years there were more differences in litter size between

mating season within farms, as well as between farms (Figure10). As all three farms were located in

the similar climatic region, differences in litter size between farms could probably be referred to the

changes in management practices. Periodical decrease in litter size could also be a consequence of

diseases in the reproductive herd. Large number of litters allows definition of season as year-month

interaction. Large difference in estimates for two adjacent months could be a sign of some unexpected

effects caused by short-term changes. Short-term changes in litter size can be a consequence of abrupt

climatic changes as well as changes in technology practicesand other unknown sources of variation.

Lukovic

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-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

A

B

C

Num

ber

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lets

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ali

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Mating season

Farm:

1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002

Figure 10: Estimates of mating season effect on number of piglets born alive per farmSlika 10: Ocene vpliva sezone pripusta na število živorojenih pujskov po farmah


4.1.6 Previous lactation length

The effect of previous lactation on litter size is describedin Figure 11. Individual dots present mean

values for the specific day of lactation period. The majorityof data for all three farms occurred on

interval between 20 and 30 days. Linear regression on previous lactation length predicted litter size

well between days 18 and 30. Outside the interval, mean valuedots were not covered well with linear

regression, but there was usually small amount of data.

9.0

9.5

10.0

10.5

11.0

11.5

12.0

10 20 30 40 50

C

B

A

Num

ber

of

pig

lets

born

ali

ve

Previous lactation length (days)

Farm:

Figure 11: Relationship between the number of piglets born alive and previous lactation length perfarmSlika 11: Povezava med dolžino predhodne laktacije in številom živorojenih pujskov po farmah

Linear regression coefficients for lactation length rangedbetween 0.026 for farm C and 0.041 for farm

B (Table 17). Prolongation of lactation for 10 days caused anincrease of litter size between 0.26 and

0.41 piglets born alive. Standard errors of estimate were similar among farms.

Table 17: Estimates of linear regression coefficient with standard errors (SEE) for lactation length perfarmTabela 17: Ocene linearnih regresijskih koeficientov in njihove standardne napake (SEE) za dolžinolaktacije po farmah

Farm Estimate SEE p-valueA 0.035 ± 0.002 <0.001B 0.042 ± 0.002 <0.001C 0.026 ± 0.002 <0.001

Although linear regression seems not to be the best choice for presentation of this effect on the whole

interval, additional analysis showed that nesting of linear regression for previous lactation length


within lactation intervals did not changed ranking of animals on breeding value basis. The whole

interval of lactation length was divided in three subintervals (1 - 18, 19 - 34, 35 - 60 days). Breeding

values obtained with repeatability model which included previous lactation length were defined two

ways (linear regression on the whole interval and linear regression nested within three intervals) and

did not differ. Spearman rank correlation coefficients between those two sets of breeding values were

close to one (0.998). Therefore, presentation of effect of lactation length as simple linear regression

along the whole interval is sufficient from practical point of view.

4.1.7 Sow genotype

Differences between sow genotypes were presented as deviations from the Swedish Landrace (Ta-

ble 18). The largest difference between two genotypes was observed on farm C, where Large White

sows had 0.72 piglets per litter less than Swedish Landrace sows. The smallest litter size was observed

for Large White sows on all three farms.

Table 18: Comparison among sow genotypes on three farmsTabela 18: Ocene razlik* med genotipi svinj s standardnimi napakami (SEE) po farmah

Farm Genotype Estimate± SEE p-value HeterozisSL x LWa 0.159± 0.035 <0.0001 0.440

A LW x SLa 0.489± 0.254 0.0545 0.770LW -0.561± 0.131 <0.000112,21 vs 11,22** 0.605± 0.131 <0.0001SL x LWa 0.452± 0.050 <0.0001 0.611

B LW x SLa 0.502± 0.067 <0.0001 0.660LW -0.318± 0.033 <0.000112,21 vs 11,22** 0.636± 0.053 <0.0001SL x LWa 0.003± 0.030 0.9215 0.365

C LW -0.724± 0.046 <0.000112 vs 11,22** 0.359± 0.036 <0.0001

* in relation to genotype SL; ** differences between crossbred and purebred animals;a- sow genotype given first

Crossbred sows had larger litters than purebreds, as expected. Crossbred sows had between 0.36

(farm C) and 0.64 (farm B) more piglets born alive per litter than mean of purebred genotypes. This

means between 3 to 7 % of heterozis in relation to the mean of purebreds. Excluding the effect of

sow genotype from the model showed the smallest decrease in coefficient of determination. This

could be partly explained as a consequence of small differences in reproductive performance between

genotypes used in the analyses.


4.1.8 Random effects

Random part of the model was determined using repeatabilitymodel. The simplest model had only di-

rect additive genetic effect included, while more complex models were made including other random

effects (common litter environmental, permanent environmental, maternal genetic effect). Previous

analyses with different random effects in the model showed neglectable magnitude of estimates for

direct maternal effect (Table 19). The importance of maternal effect in our populations was verified

by including direct additive genetic effect, maternal additive genetic effect, and permanent environ-

mental effect. Common litter environmental effect was omitted from the model because it confounded

with maternal effect.

Table 19: Estimates of (co)variance matrices with standarderrors of estimate in the model withmaternal additive genetic effect on three farmsTabela 19: Ocene matrik kovarianc sa standardnimi napakamiocen v modelu z maternalnim aditivnimgenetskim vplivom

Farm Effect Variance Ratio Cov (am) Corr (am)Direct additive gen. 0.804± 0.015 0.103± 0.005 0.006± 0.005 0.033± 0.009

A Maternal additive gen. 0.037± 0.004 0.005± 0.001Permanent environ. 0.520± 0.012 0.067± 0.004Direct additive gen. 0.856± 0.011 0.109± 0.004 0.021± 0.003 0.999± 0.005

B Maternal additive gen. 0.020± 0.001 0.001± 0.001Permanent environ. 0.561± 0.009 0.070± 0.003Direct additive gen. 0.693± 0.011 0.105± 0.004 -0.004± 0.003 -0.101± 0.025

C Maternal additive gen. 0.003± 0.002 0.000± 0.001Permanent environ. 0.440± 0.008 0.066± 0.003

Cov (am) - covariance between direct and maternal additive genetic effects; Corr (am) - correlation between direct and

maternal additive genetic effects

Estimated genetic correlations between the direct additive and maternal additive genetic effects were

negative and varied from -0.23 to -0.87 in the previous studyat one farm in Slovenia (Sadek-Pucnik

and Kovac, 1996). These results are in agreement with the recent results of Chen et al. (2003). Based

on the previous study, maternal genetic effect estimation and genetic correlation with direct additive

genetic effect were conducted. Estimates of maternal additive genetic effect as ratio in the phenotypic

variance were neglectable and ranged from less than 0.001 for farm C to 0.005 for farm A. At the

same time, genetic correlations between direct additive and maternal additive genetic effects were

small and mainly positive (farms A and B). Small negative genetic correlation between direct and

maternal genetic effect was found only for farm C (-0.10). Therefore, maternal genetic effect was

assumed to be not significant in any data set analysed and it was excluded from further analyses.

Alfonso et al. (1997) reported very different values estimated for the correlation between direct and

maternal additive genetic effects (between 0.9 and -1.0) and interpreted them as a consequence of the

small values for the maternal heritability (0.02 maximum).Estimates of direct additive genetic effect

and permanent environmental effect were in line with literature.


4.2 COMPUTATIONAL REQUIREMENTS

The computations were conducted on a computer Compaq Alpha PC 264DP (750 MHz). In re-

peatability model only four dispersion parameters were needed for the estimation. Multiple-trait

model needed 63 unknown dispersion parameters to estimate.The number of the unknown disper-

sion parameters for random regression model varied between30 with LG1 and 51 with LG3 for DS1,

and between 64 with LG1 and 85 with LG2 for DS2. For the estimation of unknown parameters

with repeatability model the minimum computing time was needed (Table 20). Estimation of disper-

sion parameters by random regression model needed generally less time than multiple-trait analyses.

Multiple-trait analysis with similar number of equations as random regression model with cubic Leg-

endre polynomials needed approximately four times more computing time. Around twice as much

iterations were required with MTM compared to RRM with LG3 and one iteration in multiple trait

analysis needed almost twofold more time than in RRM. Randomregression model was more robust

compared to MTM, since several runs with different startingvalues were needed for MTM to obtain

global maximum. One of the advantages of RRM is that higher parities could be included in analysis,

which is not possible with MTM due to high correlations between NBA in later parities. The speed

of convergence depended substantially on the starting values, particularly in multiple-trait analysis.

Table 20: Computational requirements in repeatability model (REP), multiple-trait model (MTM)and random regression model (RRM) with different order of Legendre polynomials (LG1–LG3) forDS1 and DS2 for farm BTabela 20: Poraba racunalniških kapacitet v ponovljivostnem modelu (REP), veclastnostnem modelu(MTM) in modelu z nakljucno regresijo (RRM) z razlicnim stopnjam Legendrovih polinomov (LG1–LG3) za nize podatkov DS1 in DS2 pri farmi B

Model Data No. of No. of No. of RAM CPU timeparameters equations iterations (Mb) (hh:mm:ss)

REP DS1 4 82682 55 103 00:33:21DS2 4 87045 43 144 00:25:28

MTM DS1 63 324061 139 392 14:17:15RRM-LG1 30 163981 51 291 00:54:18RRM-LG2 DS1 39 245876 61 324 02:11:20RRM-LG3 51 327771 69 365 02:56:05RRM-LG1 64 173887 59 344 01:16:17RRM-LG2 DS2 73 260729 85 396 02:46:41RRM-LG3 85 347571 94 456 04:39:10

4.3 REPEATABILITY MODEL

Repeatability model is a frequent procedure for genetic evaluation for litter size. Assuming the com-

plete genetic correlations between litter size in subsequent parities estimates of variance components

and their ratios in respect to phenotypic variance were calculated (Table 21). Variance components


were estimated separately by farms. The estimate of (co)variances by farms did not differ consider-

ably, because of similar management used. Estimates of variance components, except for common

litter environmental variance, were the smallest for farm C. Estimates of ratios for all random effects

were more similar than the components themselves. In general, the direct additive genetic variance

as well as phenotypic variance were large enough to enable successful selection on litter size.

Table 21: Estimates of variance components and ratios of phenotypic variance for the number ofpiglets born alive by farms with repeatability model for data set DS2Tabela 21: Ocene komponent varianc in deleži fenotipske variance za število živorojenih pujskov pofarmah v ponovljivostnem modelu za niz podatkov DS2

Farm Var(a) Var(l) Var(p) Var(e) Var(ph)A 0.831± 0.012 0.150± 0.008 0.396± 0.012 6.394± 0.009 7.772B 0.823± 0.013 0.082± 0.006 0.491± 0.011 6.677± 0.008 8.072C 0.638± 0.012 0.105± 0.007 0.347± 0.011 5.572± 0.008 6.643

h2

a l2 p2 e2

A 0.107± 0.005 0.019± 0.003 0.050± 0.005 0.823± 0.003B 0.102± 0.004 0.010± 0.002 0.061± 0.003 0.827± 0.002C 0.095± 0.004 0.015± 0.002 0.051± 0.004 0.839± 0.002

Var(a) - direct additive genetic variance, Var(l) - variance of common litter environmental effect, Var(p) - permanentenvi-

ronmental variance, Var(e) - residual error variance, Var(ph) - phenotypic variance,h2

a- direct heritability,l2- proportion

of common litter environmental effect,p2- proportion of permanent environmental effect,e2- proportion of residual error

variance

Direct additive genetic variance was similar for farms A andB (0.83 and 0.82), and slightly smaller

(0.64) on farm C. The heritability estimates obtained in repeatability model ranged between 0.095

on farm C and 0.107 on farm A. These estimates confirmed a number of piglets born alive as a low

heritable trait with average estimate of heritability for litter size around 0.10. Differences between

farms were negligible, although there has been no genetic ties among farms for nearly 20 years.

Common litter environmental variances were the smallest among all random effects and ranged from

0.08 on farm B to 0.15 on farm A. Common environment in litter presented as ratio in the phenotypic

variance explained less than 2 % of variability. Small magnitude of this effect may be a consequence

of small full-sib size in the data, averaging between 1.4 and1.5 animals gained from same litter.

Additionally, over 60 % of the litters had only one breeding sow. Proportion of the phenotypic vari-

ance explained with common litter variance might be relatively small due to crossfostering that is a

frequent management practice applied on farms.

Permanent environmental effect is characteristic for individual sows. Estimates of permanent envi-

ronmental variances were approximately one-half the size of the direct additive genetic variances. It

was the largest (0.49) on farm B, where the sows had on the average more litters per sow (3.51 in

DS1 or 4.16 in DS2) than on the other two farms. The first three litters were less similar compared to

the 3rd to 5th litters. The importance of the permanent environment was higher than common litter


environment and it explained between 5 % (farms A and C) and 6 %of phenotypic variability (farm

B).

Phenotypic variance for the number of piglets born alive ranged between farms from 6.64 on farm

C to 8.07 on farm B. The same trend as for phenotypic variance was suggested by overall variance

where it was the smallest for farm C.

Residual error variances were similar for farms A and B, and smaller for farm C. Estimates of ratios

for residual component were more similar than for other random effects. On all three farms, residual

error presented between 82 and 84 % of phenotypic variability.

4.4 MULTIPLE-TRAIT ANALYSIS

Multiple-trait analysis is preferred in a situation where genetic correlations between successive litter

sizes are substantially lower than one. Litter size in different parities are treated as different traits.

Multiple-trait analysis provides correlations among parities included for all random effects. Only

data set DS1 was used in multiple-trait analysis because of the numerical problems with the inclusion

of more parities. Litter sizes in higher parities were highly correlated and the system become not

positive definite. Although, multiple-trait analysis was not done for data set DS2, on the basis of high

genetic correlations in the last parities for data set DS1, assumption is that between NBA in the last

parities of DS2 very high genetic correlations exist.

4.4.1 Variance components

Estimates of variance components and ratios of random effects in the phenotypic variance obtained

from multiple-trait analysis (Table 22) did not differ greatly from estimates obtained with the re-

peatability model. Estimates of phenotypic variance in multiple-trait analysis increased by parity and

estimates from the repeatability model were within those estimates. Differences in estimates of the

phenotypic variance by parities among farms were a consequence of age at farrowing and variability

of age. Sows on farm A farrowed at higher age with relatively small variability of age at farrowing

and therefore difference between estimates of phenotypic variance by parities were the smallest. On

farm B, sows farrowed at similar age as sows on farm A, but variability of age at farrowing was higher.

Considerably smaller estimates of phenotypic variance on farm C was a consequence of lower age at

farrowing while variability of age at farrowing was satisfactorily.

Direct additive genetic variances were mostly higher than in repeatability model. They increased

by parities, with the exception of the last parity. The largest difference in estimates of direct additive

genetic variance by parities were noticed on farm C. Direct additive genetic variance for NBA showed

a small reduction from the first to the second parity on farms Aand C, and from the second to the third

parity on farm B, while residual variance increased. After that, variances on all three farms increased,

although total changes of direct additive variance were small. Litter size differs from growth and milk

traits where the variance change much more along the trajectory (Schaeffer, 2004). Slight decreases


in the direct additive genetic variances were observed after the fifth parity on all farms. On farm C,

estimates of direct additive genetic variance were smallerthan on other two farms, similarly to the

usage of repeatability model.

Variance of common litter environmental effect showed a decrease in the third and/or the fourth parity.

In comparison with direct additive genetic variance, magnitude of common litter environment vari-

ance was smaller. It was approximately one-third or less of the size of direct additive genetic effect.

Comparing the estimate of common litter environmental variance in repeatability model, estimates

obtained with multiple-trait analysis were higher.

Residual error variances changed over parities in the relatively narrow range on all three farms. They

ranged from 6.2 to 6.9 on farm A, from 6.2 to 7.2 on farm B, and from 5.1 to 6.0 on farm C. Estimates

of residual variance differed a little among farms, and increased slightly by parities. Similar as for

phenotypic variance, the smallest residual variance was found on farm C. Estimates were also slightly

higher than estimates obtained with repeatability model.

Heritability estimates for the number of piglets born alivechanged by parities and generally ranged

between 0.10 and 0.14 (Table 22). They were also low, but a little higher than in repeatability model.

Heritabilities increased up to the fifth parity, and reducedfor the sixth parity, what coincide with the

reduction of litter size after the fifth parity. The deviation of heritability was observed in the second

parity. In general, heritability estimates obtained with multiple-trait model were higher than those

gained from univariate repeatability analysis.

The ratio for the common litter environmental variance withrespect to the total variance ranged

between 0.01 and 0.06. These estimates were higher than those obtained in repeatability model (below

0.02). The smallest estimates were obtained in the middle oftrajectory (the third and fourth parity).

The small magnitude of this effect may be due to the relatively large amount of crossfostering that

was practiced on farms in Slovenia, although information onthe exact number of piglets nursed or

litters affected was not provided. The highest estimates ofthis effect were in the sixth parity for all

three farms.

In the multiple-trait analysis, permanent environmental effect within animal can not be separated from

the residual. Estimates of residual error variances as ratios by parities were similar between the farms

and ranged between 0.82 and 0.88. Estimates of residual error were similar to estimates obtained in

repeatability model. Standard errors for all estimates of variances and ratios were lower than 0.05.

Lukovic

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forlitter

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arandom

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Ljubljana,U

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Table 22: Estimated variance components with standard errors of estimates in multiple-trait modelTabela 22: Ocene komponent variance in standardne napake ocen v veclastnostnem modelu

Farm Parity Var(a) Var(l) Var(e) Var(ph) h2

a l2 e2

1 0.861± 0.022 0.347± 0.017 6.241± 0.025 7.449 0.115± 0.008 0.046± 0.005 0.837± 0.0092 0.824± 0.022 0.290± 0.023 6.948± 0.033 8.062 0.102± 0.007 0.036± 0.007 0.861± 0.010

A 3 0.889± 0.021 0.120± 0.012 6.749± 0.029 7.758 0.114± 0.007 0.015± 0.004 0.869± 0.0084 1.127± 0.030 0.104± 0.013 6.517± 0.034 7.748 0.145± 0.010 0.013± 0.004 0.840± 0.0105 1.032± 0.033 0.311± 0.042 6.527± 0.048 7.870 0.131± 0.011 0.039± 0.014 0.829± 0.0156 0.993± 0.029 0.393± 0.034 6.785± 0.059 8.171 0.121± 0.009 0.048± 0.011 0.830± 0.012

1 0.801± 0.016 0.165± 0.013 6.890± 0.020 7.856 0.102± 0.005 0.021± 0.004 0.876± 0.0072 0.962± 0.018 0.221± 0.019 6.198± 0.025 7.381 0.130± 0.006 0.030± 0.006 0.839± 0.008

B 3 0.903± 0.018 0.057± 0.007 7.031± 0.024 7.991 0.112± 0.006 0.007± 0.002 0.879± 0.0064 0.915± 0.019 0.219± 0.014 7.058± 0.029 8.192 0.111± 0.006 0.026± 0.004 0.861± 0.0075 1.016± 0.022 0.280± 0.020 6.960± 0.032 8.256 0.123± 0.007 0.033± 0.006 0.842± 0.0096 1.009± 0.023 0.291± 0.041 7.248± 0.039 8.548 0.118± 0.007 0.034± 0.014 0.847± 0.014

1 0.731± 0.017 0.159± 0.013 5.139± 0.019 6.029 0.121± 0.006 0.026± 0.005 0.852± 0.0082 0.680± 0.016 0.134± 0.016 6.060± 0.022 6.874 0.098± 0.006 0.019± 0.006 0.881± 0.008

C 3 0.771± 0.017 0.178± 0.018 5.925± 0.024 6.874 0.112± 0.006 0.025± 0.007 0.861± 0.0094 0.921± 0.022 0.066± 0.008 6.003± 0.024 6.990 0.131± 0.008 0.009± 0.003 0.858± 0.0085 1.038± 0.026 0.204± 0.015 5.855± 0.028 7.097 0.141± 0.009 0.028± 0.005 0.824± 0.0096 0.757± 0.019 0.497± 0.047 5.860± 0.038 7.114 0.106± 0.007 0.069± 0.016 0.823± 0.014

Var(a) - direct additive genetic variance, Var(l) - variance of common litter environmental effect, Var(e) - residual error variance, Var(ph) - phenotypic variance,h2

a - direct heritability,

l2 - proportion of common litter environmental effect,e2 - proportion of residual error variance


4.4.2 Correlations

Genetic as well as phenotypic correlations were the highestbetween adjacent parities and decreased

as the interval between parities increased (Table 23). At the same time, correlations between pairs of

adjacent parities increased as parities increased. Low genetic correlations between NBA in the first

and the second or later parities suggest use of multivariateanalysis instead of a simple repeatability

model which assumes homogeneity of variance across parities. Genetic correlations between the

fifth and the sixth parity were close to one, and if higher parities were included in data set, high

genetic correlations between higher parities would be expected too. Standard errors for all correlation

estimates were lower than 0.05.

Table 23: Estimates of direct additive genetic (above diagonal) and phenotypic correlations (belowdiagonal) by multiple-trait model for data set DS1Tabela 23: Ocene direktnih aditivnih genetskih (nad diagonalo) in fenotipskih korelacij (pod diago-nalo) v veclastnostnem modelu za niz podatkov DS1

Farm Parity Number Parityof records 1 2 3 4 5 6

1 17544 0.858 0.738 0.643 0.501 0.5342 12370 0.172 0.946 0.918 0.809 0.875

A 3 10351 0.156 0.190 0.985 0.902 0.9484 8698 0.151 0.164 0.222 0.903 0.9775 6990 0.111 0.162 0.206 0.219 0.9546 5462 0.115 0.148 0.197 0.227 0.204

1 27243 0.800 0.719 0.710 0.662 0.6802 19229 0.178 0.887 0.869 0.852 0.854

B 3 17167 0.167 0.191 0.882 0.896 0.8844 14392 0.133 0.207 0.200 0.950 0.9495 11907 0.126 0.176 0.198 0.211 0.9996 9574 0.116 0.149 0.192 0.213 0.222

1 26809 0.856 0.859 0.793 0.743 0.7692 18185 0.158 0.958 0.891 0.827 0.792

C 3 14326 0.166 0.178 0.982 0.951 0.9324 11139 0.149 0.177 0.210 0.992 0.9795 8808 0.144 0.157 0.234 0.218 0.9926 6719 0.123 0.147 0.215 0.204 0.197

Genetic correlations between the first and later parities generally ranged between 0.85 to 0.53. Direct

additive genetic correlations between the first and the second parity ranged between 0.80 and 0.86.

The number of piglets born alive in the first parity could be genetically a different trait than litter size

in later parities. Sometimes, NBA in the second parity couldbe regarded as a different trait too, when

sows were in the poor condition after the first parity. Such problem may be present on farm B, where

genetic correlations between the second and third parity were under 0.90. Genetic correlations above

0.90 were found in our study after the fourth parity in farm B,but for farm A and C very high genetic


correlations (>0.94) were found already between the secondand the third parity. Partly, lower genetic

correlations between the first and higher parities could be explained by selection on high meatiness

that might cause some problems in sow reproduction, especially in the first parities.

Estimates of phenotypic correlations (Table 23) were much lower than the genetic correlations (0.11

- 0.22). However, the pattern of the highest values between adjacent parities and decrease between

more distant parities was similar for genetic and phenotypic correlations. The phenotypic correlations

increased as a pair of adjacent parities increased.

Correlation estimates for common litter environment were in magnitude between the direct additive

genetic correlations and phenotypic correlations. They oscillated between pairs of adjacent parities

(Table 24), and increased linearly over parities similar todirect additive genetic correlations. Oscilla-

tion in common litter environment correlations were observed on all farms.

Residual correlations were definitively the smallest of allcorrelations and were mainly smaller than

0.10 (Table 24). Similarly, as for direct additive genetic and phenotypic correlations, residual corre-

lations showed increasing tendency when pairs of adjacent parities increased.

Table 24: Estimates of common litter environmental (above diagonal) and residual (below diagonal)correlations by multiple-trait model for data set DS1Tabela 24: Ocene korelacij za skupno okolje v gnezdu (nad diagonalo) in za ostanek (pod diagonalo)v veclastnostnem modelu za niz podatkov DS1

Farm Parity Number Parityof records 1 2 3 4 5 6

1 17544 0.493 0.771 0.757 0.677 0.5422 12370 0.068 0.932 0.067 0.906 0.243

A 3 10351 0.059 0.076 0.380 0.951 0.0664 8698 0.058 0.060 0.105 0.463 0.9485 6990 0.025 0.039 0.085 0.100 0.1796 5462 0.031 0.073 0.099 0.088 0.091

1 27243 0.631 0.068 0.666 0.105 0.1622 19229 0.068 0.816 0.076 0.362 0.186

B 3 17167 0.086 0.075 0.583 0.405 0.3664 14392 0.034 0.115 0.114 0.115 0.2655 11907 0.048 0.066 0.095 0.108 0.1556 9574 0.044 0.042 0.093 0.126 0.114

1 26809 0.471 0.531 0.534 0.059 0.4942 18185 0.062 0.987 0.872 0.834 0.983

C 3 14326 0.061 0.063 0.934 0.812 0.9974 11139 0.047 0.073 0.088 0.744 0.9405 8808 0.056 0.044 0.106 0.081 0.8376 6719 0.018 0.034 0.083 0.076 0.043


4.5 RANDOM REGRESSION MODEL

Random regression model included fixed and random part. Fixed part was the same as in the repeata-

bility model. Random part of the RRM included direct additive genetic, common litter environmental

and permanent environmental effect. Random effects were fitted as random regressions on Legendre

polynomials of different power.

4.5.1 Eigenvalues and eigenfunctions

Eigenvalues for Legendre polynomials from linear to cubic power for all random effects were com-

puted (Table 25). The eigenvalues of genetic and environmental covariance matrices of random re-

gression coefficients quantify the relative importance of each order of Legendre polynomials. The

sum of eigenvalues for direct additive genetic and permanent environmental variance increased with

the power of Legendre polynomials. On the other hand, the sumof eigenvalues for common litter

effect was on the same level regardless the model. The first eigenvalue increased for all random ef-

fects in all data sets with increasing order of Legendre polynomials, but decreased as the a proportion

of total variability. The first three eigenvalues explainedmore than 99 % for direct additive genetic

and common litter environmental variance. For permanent environmental effect, only the first two

eigenvalues could be enough to explain most of variability.Although, equal power of Legendre poly-

nomials was considered in order to provide equal opportunity of the variation for all random effects,

some evidence has been found in favor to a lower order needed for some random effects (van der Werf

et al., 1998).

The eigenvalues of genetic covariance functions for DS1 (Table 25) showed that the constant (zero)

term accounted between 90 and 95 % of the additive genetic variability for NBA. Thus, approximately

5 to 10 % of variability was explained by individual genetic curve of a sow. The constant term for

permanent environmental effect in data set DS1 with six parities ranged between 91 % on farm B and

97 % on farm A. Smaller proportion for the constant term was noticed for common litter environmen-

tal effect which ranged from 60 % on farm B to 92 % on farm C. The first eigenvalue was the largest

for common litter environmental effect. Although, part of variability explained with individual com-

mon litter environmental curve of a sows was the largest among random effects, the relevance of them

is small due to small variance estimates. Eigenvalues of covariance functions showed that quadratic

Legendre polynomials with three regression coefficients sufficed to model permanent environment

effect.

Extending data to the tenth parity caused a decrease in explained proportion of variance by the zero-

th eigenvalue for direct additive genetic effect on farms. The constant (zero-th term) explained from

90 % in LG1 on farm C to 85 % in LG3 on farm A (Table 26). The rest ofvariability (between 10

and 15 %) was explained by individual sow curves. This percentage of the genetic variability could

be interesting for selection on the shape of production curve for litter size. Therefore, selection on

persistency is interesting only if data from higher parities are used. The result is expected, maximum

litter size is reached in the fourth or fifth parity. The decrease of production is not clearly shown


Table 25: Eigenvalues of estimated covariance matrices of random-regression coefficients with pro-portion (in parenthesis) of the total variability for random effects in data sets DS1 with different orderof Legendre polynomials (LG1 – LG3)Tabela 25: Lastne vrednosti za ocenjene matrike kovarianc za nakljucne regresijske koeficiente zdeležem (v oklepajih) v celotni varianci za nakljucne vplive v nizih podatkov DS1 za razlicne stopnjeLegendrovih polinomov (LG1 – LG3)

Model Eigenvalue

Farm Effect order 0th 1st 2nd 3rd Sum

LG[1] 1.678 (89.97) 0.187 (10.03) 1.865

GEN LG[2] 1.745 (91.02) 0.162 (8.45) 0.010 (0.52) 1.917

LG[3] 1.743 (90.03) 0.171 (8.83) 0.013 (0.67) 0.009 (0.46) 1.936

LG[1] 1.009 (97.11) 0.030 (2.89) 1.039

A PERM LG[2] 1.073 (93.79) 0.071 (6.21) 0.000 (0.00) 1.144

LG[3] 1.081 (91.92) 0.095 (8.08) 0.000 (0.00) 0.000 (0.00) 1.176

LG[1] 0.243 (89.01) 0.030 (10.99) 0.273

LITT LG[2] 0.226 (87.94) 0.031 (12.06) 0.000 (0.00) 0.257

LG[3] 0.221 (82.16) 0.048 (17.84) 0.000 (0.00) 0.000 (0.00)0.269

LG[1] 1.732 (95.06) 0.090 (4.94) 1.822

GEN LG[2] 1.798 (94.68) 0.084 (4.42) 0.017 (0.89) 1.899

LG[3] 1.790 (93.91) 0.100 (5.10) 0.020 (0.98) 0.000 (0.00) 1.910

LG[1] 1.167 (91.45) 0.109 (8.54) 1.276

B PERM LG[2] 1.215 (91.14) 0.118 (8.85) 0.000 (0.00) 1.333

LG[3] 1.222 (90.85) 0.120 (9.14) 0.000 (0.00) 0.000 (0.00) 1.342

LG[1] 0.109 (66.06) 0.056 (33.93) 0.165

LITT LG[2] 0.100 (63.29) 0.058 (36.71) 0.000 (0.00) 0.158

LG[3] 0.100 (60.24) 0.057 (34.33) 0.009 (5.42) 0.000 (0.00)0.166

LG[1] 1.503 (94.77) 0.083 (5.23) 1.586

GEN LG[2] 1.558 (94.36) 0.069 (4.18) 0.024 (1.46) 1.651

LG[3] 1.584 (93.01) 0.094 (5.52) 0.025 (1.47) 0.000 (0.00) 1.703

LG[1] 0.759 (96.19) 0.030 (3.81) 0.789

C PERM LG[2] 0.827 (94.08) 0.052 (5.92) 0.000 (0.00) 0.879

LG[3] 0.833 (93.38) 0.059 (6.62) 0.000 (0.00) 0.000 (0.00) 0.892

LG[1] 0.274 (90.43) 0.029 (9.57) 0.303

LITT LG[2] 0.270 (92.15) 0.023 (7.85) 0.000 (0.00) 0.293

LG[3] 0.270 (92.78) 0.021 (7.22) 0.000 (0.00) 0.000 (0.00) 0.291

GEN - direct additive genetic effect, PERM - permanent environmental effect, LITT - common litter environmental effect

when only the first six parities were considered. By adding more parities to data set, the decrease in

production could be seen on raw, as well as on analyzed data. Besides persistency i.e. decrease after

peak production, an increase of production curve could be selected for.

Similar to direct additive genetic effect, the proportion of variance explained by the constant term

for permanent environmental effect decreased by prolongation of data set to the tenth parity. The

decrease was larger compared to the decrease for direct additive genetic effect. The constant term for

permanent environmental variance explained between 70 and95 % of variability what means that up


to 30 % of permanent environmental variability was explained by deviation of individual curves of

sows. Constant term for the common litter environmental effect oscillated with increased power of

Legendre polynomials and differed between farms. But, due to a small magnitude of this effect, these

changes could be of minor importance.

Table 26: Eigenvalues of estimated covariance matrices of random regression coefficients with pro-portion (in parenthesis) of the total variability for random effects in data sets DS2 with different orderof Legendre polynomials (LG1–LG3)Tabela 26: Lastne vrednosti za ocenjene matrike kovarianc za nakljucne regresijske koeficiente zdeležem (v oklepajih) v celotni varianci za nakljucne vplive v nizu podatkih DS2 za razlicne stopnjeLegendrovih polinomov (LG1–LG3)

Model Eigenvalue

Farm Effect order 0th 1st 2nd 3rd Sum

LG[1] 2.069 (87.00) 0.309 (13.00) 2.378

GEN LG[2] 1.710 (82.05) 0.296 (14.20) 0.078 (3.75) 2.084

LG[3] 1.708 (85.01) 0.255 (12.69) 0.041 (2.04) 0.005 (0.25)2.009

LG[1] 1.109 (93.19) 0.081 (6.81) 1.190

A PERM LG[2] 0.979 (69.87) 0.421 (30.04) 0.001 (0.07) 1.401

LG[3] 1.064 (70.37) 0.429 (28.37) 0.019 (1.26) 0.000 (0.00)1.512

LG[1] 0.336 (81.55) 0.076 (18.45) 0.412

LITT LG[2] 0.426 (91.61) 0.039 (8.39) 0.000 (0.00) 0.465

LG[3] 0.367 (90.39) 0.038 (9.36) 0.001 (0.25) 0.000 (0.00) 0.406

LG[1] 2.024 (90.23) 0.219 (9.77) 2.243

GEN LG[2] 1.898 (85.96) 0.262 (11.86) 0.048 (2.18) 2.208

LG[3] 1.897 (85.79) 0.258 (11.67) 0.038 (1.72) 0.018 (0.82)2.211

LG[1] 1.260 (83.11) 0.256 (16.89) 1.516

B PERM LG[2] 1.159 (71.36) 0.438 (26.97) 0.027 (1.67) 1.624

LG[3] 1.150 (72.46) 0.404 (25.46) 0.033 (2.08) 0.000 (0.00)1.587

LG[1] 0.214 (78.67) 0.058 (21.33) 0.272

LITT LG[2] 0.233 (84.11) 0.037 (13.36) 0.007 (2.53) 0.277

LG[3] 0.251 (76.76) 0.041 (12.54) 0.035 (10.70) 0.000 (0.00) 0.327

LG[1] 1.587 (90.84) 0.160 (9.16) 1.747

GEN LG[2] 1.505 (90.66) 1.127 (7.65) 0.028 (1.69) 1.660

LG[3] 1.500 (89.93) 0.143 (8.57) 0.025 (1.50) 0.000 (0.00) 1.668

LG[1] 0.839 (94.80) 0.046 (5.20) 0.885

C PERM LG[2] 0.797 (83.45) 0.158 (16.54) 0.001 0.08 0.955

LG[3] 0.780 (82.89) 0.151 (16.05) 0.010 (1.06) 0.000 (0.00)0.941

LG[1] 0.181 (77.68) 0.052 (22.32) 0.233

LITT LG[2] 0.186 (78.81) 0.050 (21.19) 0.000 (0.00) 0.236

LG[3] 0.195 (75.87) 0.062 (24.13) 0.000 (0.00) 0.000 (0.00)0.257

GEN - direct additive genetic effect, PERM - permanent environmental effect, LITT - common litter environmental effect

The proportion of direct additive genetic variance for higher terms was mainly covered by linear

and quadratic coefficients. For data sets DS1 per farm linearcoefficients accounted for 4.18 % to

10.03 %; for data sets DS2 linear coefficient ranged between 7.65 % and 14.20 % per farm. Quadratic


-3

-2

-1

0

1

2

3

1 2 3 4 5 6 7 8 9 10

EF1EF2EF3EF4 ZERO

Nu

mb

er o

f p

igle

ts b

orn

ali

ve

Parity

Eigenfunction:

Figure 12: Estimated genetic eigenfunctions from random regression model with cubic power fornumber of piglets born alive on farm BSlika 12: Ocenjene genetske lastne funkcije za Legendrov polinom tretje stopnje v modelu znakljucno regresijo za število živorojenih pujskov na farmi B

coefficients covered direct additive genetic variance between 0.52 % and 1.47 % per farm in data

sets DS1. In data sets DS2 the proportions of direct additivegenetic variance covered by quadratic

coefficients were higher than in data sets DS1 and ranged between 1.50 and 3.75 % per farm. The

contribution of cubic coefficients in the variability of random effects was negligible.

The genetic eigenfunctions in RRM with cubic Legendre polynomials for farm B were estimated

(Figure 12). The first eigenfunction (EF1) explained between 85 % and 90 % of the genetic variance

for the number of piglets born alive. It was positive and moreor less on the same level from the first

to the tenth parity. Thus, selection for the level of litter size at any parity causes a similar response

for all parities. The size of the first eigenvalue indicated that selection on the first component would

produce rapid changes in the observed trait. The second eigenfunction which explained between 7 %

and 14 % of the genetic variance changed the sign after the fifth parity. Apparently, genetic changes

in the slope curve from the first to the fifth parity would causethe decrease in litter size after the

fifth parity, and vice-versa. The size of the second eigenvalue indicated that the response to selection

involving the second eigenfunction would be slower than forchanges involving the first eigenfunction.

Magnitudes of the third and the fourth eigenvalue were negligible. Therefore, the change of the slope

of eigenfunctions was of little relevance.

4.5.2 Covariance components

Estimated variance components and proportions in the phenotypic variance in random regression

analysis for data set DS1 (Table 27) were similar to the estimates obtained by multiple-trait analysis

on all three farms. The estimated direct additive genetic variance for the number of piglets born alive


changed per parity in random regression analysis for all three farms. The estimates of direct additive

genetic variance ranged between 0.70 and 1.05.

Table 27: Estimated variance components and proportions inthe phenotypic variation for NBA inrandom regression model with cubic Legendre polynomial forthree farms in data set DS1Tabela 27: Ocene komponent variance in deleži v fenotipski varianci za model z nakljucno regresijoz uporabljenim Legendrovim polinomom tretje stopnje po farmah za niz podatkov DS1

Farm Parity Var(a) Var(l) Var(p) Var(e) Var(ph) h2

a l2 p2 e2

1 0.90 0.33 0.33 5.88 7.44 0.121 0.044 0.045 0.790

2 0.88 0.13 0.53 6.52 8.06 0.109 0.016 0.066 0.809

A 3 0.96 0.06 0.64 6.09 7.75 0.124 0.008 0.082 0.785

4 1.03 0.08 0.69 5.85 7.66 0.135 0.011 0.090 0.764

5 1.05 0.16 0.65 6.05 7.90 0.132 0.020 0.082 0.766

6 0.97 0.21 0.47 6.46 8.10 0.120 0.026 0.057 0.797

1 0.79 0.17 0.42 6.41 7.79 0.101 0.022 0.054 0.823

2 0.94 0.08 0.52 5.81 7.35 0.128 0.011 0.071 0.790

B 3 1.01 0.04 0.63 6.26 7.94 0.127 0.005 0.080 0.789

4 0.99 0.05 0.72 6.32 8.09 0.122 0.007 0.090 0.781

5 1.00 0.09 0.79 6.27 8.15 0.122 0.011 0.097 0.770

6 1.04 0.16 0.84 6.40 8.43 0.123 0.019 0.099 0.758

1 0.72 0.12 0.41 4.73 5.99 0.121 0.021 0.069 0.789

2 0.67 0.17 0.35 5.73 6.91 0.097 0.024 0.050 0.828

C 3 0.81 0.14 0.46 5.40 6.82 0.119 0.020 0.068 0.792

4 0.99 0.09 0.55 5.38 7.00 0.141 0.013 0.078 0.768

5 1.05 0.11 0.50 5.42 7.08 0.148 0.016 0.071 0.765

6 0.72 0.42 0.34 5.58 7.05 0.101 0.059 0.049 0.791

Var(a) - direct additive genetic variance, Var(l) - variance of common litter environmental effect, Var(e) - residual error

variance, Var(ph) - phenotypic variance,h2

a - direct heritability,p2- proportion of permanent environmental effect,l2 -

proportion of common litter environmental effect,e2 - proportion of residual error variance

Common litter environmental variance estimates ranged between 0.06 and 0.33 for farm A, for farm

B between 0.04 and 0.17, and between 0.09 and 0.42 for farm C. Estimates of common litter environ-

mental variance were similar to those obtained by multiple-trait model. Estimates of common litter

environmental variance showed a similar tendency of the lowest estimates in the third or fourth parity

as estimates from multiple-trait model.

Multiple-trait analysis did not allow inclusion of permanent environmental effect in the model, but

in the RRM analysis permanent environmental effect was fitted as random regression on parity. The

estimates of permanent environmental variance changed over parities and ranged between 0.33 and

0.69 for farm A, between 0.42 and 0.84 for farm B, and between 0.35 and 0.55 for farm C. Estimates

of permanent environmental variance obtained with repeatability model ranged between 0.39 and 0.49

and were in agreement with those found in RRM analysis.

Estimates of residual error variance by parity ranged between 5.85 and 6.46 for farm A, for farm

B between 5.81 and 6.41, and between 4.73 and 5.73 for farm C. These estimates were smaller


than estimates of residual error variance in multiple-trait analysis because in multiple-trait analysis

permanent environment variance was included in residual error variance. Estimates of phenotypic

variance agreed in RRM and MTM analysis. Generally, changesin the phenotypic variance were

small and they oscillated a little by parities. On farm C somesmaller estimates of phenotypic variance

were found in agreement with estimates for farm C in the MTM analysis.

Litter size analyzed as the number of piglets born alive withrandom regression model had low her-

itability, but was slightly higher than estimates obtainedby multiple-trait analysis. Heritability esti-

mates ranged between 0.10 and 0.13 by parities for farms A andB, and between 0.09 and 0.15 for

farm C. Heritability estimates by farm obtained using a repeatability model was mainly lower than

estimates in multiple-trait analysis, and were within these ranges. Increasing tendency of heritability

estimates by parities was in agreement with multiple-traitanalysis.

The ratios for the common litter environmental effect with respect to the total variance were low.

They ranged between 0.008 and 0.044 for farm A, between 0.054and 0.099 for farm B, and between

0.050 and 0.078 for farm C (Table 27). The ratios of permanentenvironmental effect with respect to

the total variance ranged between 0.045 and 0.090 for farm A,between 0.054 and 0.099 for farm B,

and between 0.049 and 0.078 for farm C. They were similar to the estimates found in the repeatability

model which were in the range from 0.05 to 0.06. The smallest estimates of common litter environ-

mental effect were in the third and the fourth parity and was in agreement with the estimates from

MTM analysis. The ratios for residual variance was similar among farms, and ranged between 0.76

and 0.82 per parity.

Magnitudes of estimated ratio in the phenotypic variance for random effects were similar among

farms (Figure 13). Estimates of ratio in phenotypic variance for direct additive genetic effect ranged

between 10 and 15 %, for permanent environmental effect between 5 and 10 %, and the common litter

environmental effect explained less than 2 % of phenotypic variance. The estimated ratios for direct

additive genetic and common litter environmental effect were in agreement with those obtained using

a multiple-trait model.

The estimates of variance components and proportions in thephenotypic variance in data sets DS2

(Table 28) were similar to the estimates obtained in data sets DS1. The estimates of direct additive

genetic variance showed a difference between farms. On farmA, estimates of direct additive genetic

variance from the first to the ninth parity ranged between 0.92 and 1.08, and in the tenth parity

decreased to 0.77. On farm B, direct additive variance increased gradually from 0.79 in the first

parity to 1.27 in the tenth parity and this was the largest change in the direct additive variance among

farms. On farm C, direct additive genetic variance increased from 0.67 in the first parity to 0.91

in the eighth, and then decreased to 0.81 in the tenth parity.Estimates of permanent environmental

variance showed similar increasing tendency along trajectory, especially in the last parities. Estimates

of common litter environmental variance showed the lowest values in the third and the fourth parity

on farms B and C, while on farm C the magnitude of common littervariances by parities was more or

less on the same level. Larger change in a common litter variance was observed from the ninth to the

tenth parity.


0.00

0.05

0.10

0.15

0.20

1 2 3 4 5 6

Additive genetic effect

Permanent environment

Common litter environment

Est

imat

ed r

atio

in

th

e p

hen

oty

pic

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ian

ce

Parity

Farm A

0.00

0.05

0.10

0.15

0.20

1 2 3 4 5 6




Est

imat

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atio

in

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ian

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Farm B

0.00

0.05

0.10

0.15

0.20

1 2 3 4 5 6




Est

imat

ed r

atio

in t

he

phen

oty

pic

var

iance

Parity

Farm C

Figure 13: Comparison of ratios in the phenotypic variance with random regression model (lines) andmultiple-trait model (triangles) for data set DS1 over parities per farmSlika 13: Primerjave deležev od fenotipske variance v modelu z nakljucno regresijo (crte) in v veclast-nostnem modelu (trikotniki) za niz podatkov DS1 po zaporednih prasitvah po farmah


Phenotypic variance of the number of piglets born alive increased from the first to the tenth parity

(Figure 14). But these changes were relatively small. The phenotypic variance differed between

farms, on farm C it was of smaller magnitude compared to estimates obtained for the other two farms.

The estimates of heritability for the number of piglets bornalive by parity ranged between 0.10

and 0.14 and was in agreement with the results of other studies (Haley et al., 1988; Rothschild and

Bidanel, 1998). The estimates of direct additive genetic effect with respect to the phenotypic variance

differed between farm B on the one side and farms A and C on the other. On farm B, estimates of the

heritability increased gradually from the first to the tenthparity. On farms A and C, the heritability by

parity increased up to the eighth parity, with the exceptionof the second parity, where a characteristic

heritability decrease was determined. At the end of trajectory the decrease in heritability was noticed

after the eighth parity. This decrease was a consequence of selection in the previous parities and

reduced amount of data in the late parities in data sets DS2.


Table 28: Estimated variance components and proportions ofphenotypic variation for NBA in randomregression model with cubic Legendre polynomial for the three farms in data set DS2Tabela 28: Ocene komponent variance in deleži od fenotipskevariance v modelu z nakljucno regresijoz uporabljenim Legendrovim polinomom tretje stopnje po farmah za niz podatkov DS2

Farm Parity Var(a) Var(p) Var(l) Var(e) Var(ph) h2

a p2 l2 e2

1 0.94 0.33 0.27 5.86 7.41 0.127 0.044 0.037 0.790

2 0.94 0.44 0.10 6.40 7.87 0.119 0.055 0.013 0.819

3 1.03 0.57 0.05 6.08 7.72 0.134 0.074 0.006 0.787

4 1.08 0.62 0.06 5.85 7.61 0.142 0.081 0.008 0.769

A 5 1.08 0.58 0.14 6.09 7.89 0.137 0.073 0.018 0.772

6 1.06 0.51 0.27 6.28 8.13 0.131 0.063 0.033 0.773

7 1.05 0.52 0.39 5.90 7.86 0.134 0.066 0.050 0.751

8 1.02 0.74 0.41 5.41 7.58 0.134 0.098 0.054 0.714

9 0.92 1.41 0.26 5.71 8.30 0.111 0.170 0.032 0.687

10 0.77 2.85 0.04 3.95 7.62 0.102 0.375 0.006 0.518

1 0.79 0.39 0.19 6.42 7.79 0.102 0.051 0.024 0.823

2 0.90 0.53 0.09 5.86 7.39 0.122 0.072 0.012 0.794

3 1.04 0.62 0.06 6.26 7.97 0.130 0.077 0.007 0.786

4 1.10 0.65 0.06 6.32 8.13 0.135 0.080 0.007 0.777

B 5 1.12 0.68 0.07 6.31 8.18 0.137 0.083 0.009 0.771

6 1.14 0.73 0.10 6.44 8.41 0.136 0.087 0.011 0.766

7 1.19 0.82 0.12 6.13 8.26 0.144 0.100 0.015 0.742

8 1.23 0.98 0.18 6.48 8.87 0.138 0.111 0.020 0.731

9 1.23 1.20 0.36 6.04 8.83 0.140 0.136 0.040 0.684

10 1.27 1.50 0.85 5.46 9.09 0.139 0.165 0.094 0.601

1 0.67 0.25 0.14 4.93 5.99 0.113 0.041 0.023 0.824

2 0.70 0.31 0.11 5.77 6.90 0.101 0.045 0.017 0.837

3 0.80 0.38 0.13 5.40 6.72 0.119 0.057 0.020 0.804

4 0.85 0.43 0.14 5.45 6.88 0.124 0.062 0.021 0.792

C 5 0.87 0.44 0.13 5.44 6.88 0.126 0.065 0.020 0.790

6 0.87 0.46 0.11 5.55 7.00 0.125 0.065 0.016 0.794

7 0.90 0.49 0.09 5.25 6.73 0.133 0.073 0.014 0.781

8 0.91 0.56 0.09 5.50 7.05 0.128 0.079 0.012 0.780

9 0.87 0.66 0.14 5.34 7.01 0.124 0.094 0.020 0.762

10 0.81 0.79 0.32 5.12 7.05 0.115 0.111 0.046 0.727

Var(a) - direct additive genetic variance, Var(l) - variance of common litter environmental effect, Var(e) - residual error

variance, Var(ph) - phenotypic variance,h2

a - direct heritability,p2- proportion of permanent environmental effect,l2 -

proportion of common litter environmental effect,e2 - proportion of residual error variance

Lukovic

Z.C

ovariancefunctions

forlitter

sizein

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odel.D


Ljubljana,U

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jubljana,Bio

technicalFaculty,Z

ootechnicalDepartm

ent,200672

5.5

6.0

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0

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1 2 3 4 5 6 7 8 9 10

ABC

Co

mm

on

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ter

effe

ct

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Farm:

Figure 14: Phenotypic variances and proportions of the phenotypic variance over parities with Legendre polynomials ofthe cubic power for the numberof piglets born aliveSlika 14: Fenotipske variance in deleži od fenotipskih varianc po zaporednih prasitvah z uporabljenim Legendrovim polinomom tretje stopnje za številoživorojenih pujskov


The estimates of permanent environmental effect expressedas a ratio generally increased over parities

and ranged from 0.05 in the first parity to 0.15 in the tenth, except for farm A where estimates of

permanent environmental effect unexpectedly increased after the eighth parity. From all random

effects, permanent environmental effect as ratio showed the smallest difference between farms. The

ratio of the common litter environmental effect was small and ranged generally from 0.0 to 0.2. Some

discrepancy was found in the last two parities.

Estimates of ratio in the phenotypic variance for both data sets showed that they were similar for the

first six parities (Figure 15). Although, the limitation of random regression model for estimates at the

end of the trajectory mentioned, random regression analysis on larger data set confirmed estimates

obtained on smaller data set. After the eighth parity estimates of ratio in the phenotypic variance

considerably changed (decrease or increase) and should be considered with caution.

4.5.3 Correlations

Correlation structure was derived from the estimated random regression coefficients for direct addi-

tive genetic effect and phenotypic variance (Table 29), as well as for common litter and permanent

environmental effect (Table 30).

Estimated genetic correlations between litter size in different parities with random regression model

showed similar tendency as the estimates obtained with multiple-trait analysis. Genetic correlations

between the number of piglets born alive at different parities were the highest between adjacent par-

ities. Direct additive genetic correlations decreased as the distance between parities increased. Esti-

mates of genetic correlations from random regression modelwere slightly higher for the majority of

the first six parities compared to the estimates obtained with multiple-trait analysis. Direct additive

genetic correlations between the first and higher parities ranged from 0.90 to 0.40 (Table 29). The

genetic correlation structure indicated that litter size at different parities is genetically not the same

trait since genetic correlations between distant measurements are much lower than unity. The genetic

correlations proved the existence of individual genetic variability, as well as justified utilization of

random regression approach in the genetic evaluation. In Figure 16 correlation structure for direct

additive genetic effect for farm B is presented as three dimensional surface plot, and was similar for

the other two farms.


0.00

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Direct additive genetic



Est

imat

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atio

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Farm A

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Common litter environmentPermanent environment

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Direct additive genetic

Common litter environment Permanent environment

Est

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Farm C

Figure 15: Comparison of estimates for ratios in the phenotypic variance between data sets DS1 andDS2 per farmSlika 15: Primerjava ocen deležev fenotipske variance med nizi podatkov DS1 in DS2 po farmah


Table 29: Estimates of direct additive genetic (above diagonal) and phenotypic correlations (belowdiagonal) by RRM for data set DS2Tabela 29: Ocene direktnih aditivnih genetskih (nad diagonalo) in fenotipskih korelacij (pod diago-nalo) v modelu z nakljucno regresijo v nizu podatkih DS2

Farm Parity Number Parity

of records 1 2 3 4 5 6 7 8 9 10

1 17544 0.911 0.778 0.672 0.589 0.515 0.450 0.402 0.382 0.398

2 12370 0.185 0.965 0.904 0.828 0.737 0.641 0.561 0.527 0.563

3 10351 0.168 0.201 0.982 0.932 0.853 0.758 0.674 0.635 0.670

4 8698 0.154 0.197 0.226 0.983 0.930 0.853 0.778 0.740 0.764

A 5 6990 0.139 0.181 0.213 0.233 0.981 0.932 0.875 0.841 0.849

6 5462 0.127 0.161 0.193 0.218 0.230 0.984 0.950 0.924 0.914

7 4085 0.117 0.144 0.174 0.203 0.224 0.240 0.990 0.974 0.950

8 2732 0.105 0.125 0.153 0.184 0.212 0.237 0.267 0.995 0.967

9 1370 0.084 0.101 0.126 0.155 0.183 0.213 0.252 0.295 0.983

10 431 0.063 0.089 0.115 0.141 0.167 0.200 0.248 0.313 0.374

1 27243 0.924 0.831 0.769 0.718 0.662 0.601 0.546 0.505 0.466

2 19229 0.186 0.978 0.938 0.878 0.794 0.698 0.616 0.569 0.556

3 17167 0.168 0.212 0.986 0.943 0.865 0.768 0.682 0.632 0.617

4 14392 0.152 0.203 0.222 0.983 0.929 0.849 0.771 0.719 0.687

B 5 11907 0.133 0.185 0.211 0.228 0.981 0.929 0.868 0.816 0.760

6 9574 0.113 0.161 0.190 0.215 0.234 0.983 0.945 0.899 0.823

7 7591 0.096 0.139 0.171 0.201 0.229 0.248 0.988 0.954 0.870

8 5443 0.079 0.114 0.144 0.177 0.210 0.237 0.265 0.986 0.913

9 3509 0.070 0.101 0.128 0.161 0.196 0.229 0.266 0.291 0.964

10 2024 0.067 0.095 0.119 0.146 0.177 0.211 0.254 0.292 0.348

1 26809 0.908 0.796 0.732 0.697 0.665 0.626 0.585 0.545 0.493

2 18185 0.162 0.974 0.937 0.892 0.831 0.759 0.700 0.678 0.699

3 14326 0.155 0.183 0.989 0.955 0.896 0.822 0.764 0.753 0.803

4 11139 0.147 0.181 0.208 0.987 0.945 0.885 0.836 0.831 0.881

C 5 8808 0.140 0.174 0.203 0.214 0.985 0.947 0.911 0.909 0.943

6 6719 0.132 0.161 0.190 0.204 0.213 0.988 0.969 0.967 0.977

7 5157 0.124 0.149 0.177 0.195 0.210 0.218 0.955 0.993 0.981

8 3832 0.108 0.131 0.158 0.178 0.196 0.209 0.225 0.999 0.974

9 2687 0.091 0.117 0.146 0.169 0.190 0.206 0.226 0.234 0.982

10 1749 0.067 0.106 0.141 0.165 0.186 0.203 0.223 0.236 0.257

Phenotypic correlations obtained with random regression model in the first six parities were similar to

those obtained by multiple-trait analysis (Table 29). In the first six parities they ranged between 0.06

and 0.24. In later parities, phenotypic correlations were higher and increased to 0.26 on farm C and

0.37 on farm A. Between litter size in the first and higher parities estimated phenotypic correlations

ranged from 0.18 (between the first and the second parity) to 0.06 (between the ninth and the tenth

parity). Again, as in multiple-trait analysis, phenotypiccorrelations between adjacent parities were

higher and they increased as pairs of parities increased.


Figure 16: Genetic correlation for the number of piglets born alive between parities using RRM forfarm BSlika 16: Genetske korelacije za število živorojenih pujskov med zaporednimi prasitvami v modelu znakljucno regresijo za farmo B

Estimates of common litter environment correlations between adjacent parities were very high (Ta-

ble 30), mainly above 0.90. Generally, on all three farms there was a tendency of decreased corre-

lations with increasing distance between parities, but thedecrease occasionally was disturbed with

some unexpected low correlations.

Estimates of correlations for permanent environmental effect between litter size in adjacent parities

were very high, mostly higher than 0.90. Between litter sizein the first and higher parities permanent

environmental correlations ranged from 0.88 (between the first and the second parity) to 0.01 (be-

tween the ninth and the tenth parity) on farm A, and showed thelargest range of correlation estimates

from all random effects. Estimates of permanent environmental correlations with random regression

can not be compared with multiple-trait analysis because permanent environmental effect was not

included in multiple-trait analysis.


Solutions for the random regression coefficients of animalsobtained using random regression model

with Legendre polynomial of the third power were used to compute the predicted breeding values for

the number of piglets born alive by parities and thus producegenetic merit functions. The predicted

breeding value curves for seven service sires with the number of progenies between 500 and 800

(Figure 17) illustrate that the genetic merit functions vary between individual animals. This means

that some service sires had a genetic predisposition for higher litter size, as well as a possibility for


Table 30: Estimates of residual common litter environment (above diagonal) and permanent environ-ment correlations (below diagonal) by RRM for data set DS2Tabela 30: Ocene korelacij za skupno okolje v gnezdu (nad diagonalo) in za permanentno okolje (poddiagonalo) v modelu z nakljucno regresijo v nizih podatkov DS2

Farm Parity Number Parity

of records 1 2 3 4 5 6 7 8 9 10

1 17544 0.990 0.998 0.830 0.650 0.588 0.579 0.595 0.641 0.833

2 12370 0.882 0.979 0.744 0.537 0.469 0.459 0.477 0.527 0.752

3 10351 0.759 0.976 0.864 0.697 0.638 0.630 0.645 0.688 0.862

4 8698 0.692 0.950 0.995 0.963 0.939 0.935 0.942 0.958 0.977

A 5 6990 0.653 0.926 0.981 0.994 0.997 0.996 0.997 0.998 0.933

6 5462 0.604 0.873 0.932 0.955 0.981 0.999 0.999 0.996 0.908

7 4085 0.483 0.720 0.779 0.814 0.869 0.949 0.999 0.996 0.906

8 2732 0.283 0.460 0.514 0.560 0.641 0.779 0.937 0.998 0.915

9 1370 0.103 0.232 0.284 0.335 0.430 0.598 0.820 0.968 0.940

10 431 0.011 0.100 0.153 0.208 0.308 0.488 0.737 0.926 0.991

1 27243 0.951 0.731 0.378 0.137 0.055 0.082 0.170 0.245 0.274

2 19229 0.964 0.901 0.625 0.405 0.333 0.377 0.466 0.506 0.490

3 17167 0.881 0.975 0.899 0.756 0.706 0.740 0.770 0.711 0.609

4 14392 0.759 0.901 0.974 0.966 0.944 0.946 0.885 0.715 0.535

B 5 11907 0.594 0.773 0.892 0.970 0.996 0.977 0.864 0.641 0.430

6 9574 0.397 0.598 0.751 0.877 0.967 0.984 0.873 0.651 0.439

7 7591 0.201 0.401 0.573 0.734 0.873 0.968 0.946 0.775 0.591

8 5443 0.041 0.219 0.390 0.567 0.740 0.883 0.972 0.938 0.821

9 3509 0.061 0.078 0.232 0.407 0.593 0.767 0.898 0.976 0.968

10 2024 0.098 0.010 0.113 0.269 0.450 0.634 0.792 0.908 0.977

1 26809 0.846 0.642 0.535 0.483 0.428 0.298 0.005 0.364 0.586

2 18185 0.931 0.952 0.903 0.876 0.845 0.761 0.538 0.190 0.063

3 14326 0.850 0.982 0.991 0.981 0.968 0.923 0.770 0.481 0.245

4 11139 0.797 0.952 0.991 0.998 0.993 0.966 0.847 0.592 0.371

C 5 8808 0.756 0.910 0.960 0.988 0.998 0.980 0.878 0.640 0.426

6 6719 0.704 0.838 0.894 0.941 0.982 0.990 0.906 0.686 0.481

7 5157 0.627 0.727 0.784 0.848 0.919 0.977 0.956 0.781 0.599

8 3832 0.526 0.590 0.645 0.723 0.819 0.913 0.979 0.930 0.807

9 2687 0.418 0.452 0.506 0.594 0.709 0.830 0.930 0.985 0.968

10 1749 0.319 0.337 0.392 0.488 0.614 0.754 0.876 0.956 0.992


maintaining litter size on high level along trajectory. There are two types of differences. Genetic

merit functions between animals differed in the level, as well as in the shape of curves.

-1

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7

Figure 17: Estimated breeding values for number of piglets born alive over parities for seven siresSlika 17: Napovedi plemenskih vrednosti za število živorojenih pujskov po zaporednih prasitvah prisedmih merjascih

Figure 17 represents few different types of curves were presented. Generally they can be arranged

in two groups. The first group includes curves that increasedin the first part of trajectory (up to the

fourth or fifth parity) and than decreased towards the end of trajectory. The second group of curves

includes the curves that decreased at first, and thereafter increased up to the last parity (or one to

two parities before). The predicted breeding value curves within both groups of curves differed in

the inflexion point i.e. parity where the curve changes direction. Although it is hard to imagine that

animal genetic potential for litter size changes with parity number, it is well known that litter size

grows with increasing parity up to parity four or five and thenreduces again. There are physiological

mechanisms that turned on and off during ageing of sows and they could be controlled genetically.

The general question is which curve of a boar is preferred forselection. Choosing service sires whose

predicted breeding value increases over parities seems possible to change the shape of curves for NBA

of sows.


8.0

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Figure 18: Phenotypic averages for the number of piglets born alive over parities for seven siresSlika 18: Fenotipska povprecja za število živorojenih pujskov po zaporednih prasitvahpri sedmihmerjascih

Phenotypic averages for the number of piglets born alive by parities for seven boars show a character-

istic increase in litter size up to the fourth parity, which later decreases (Figure 18). From the fourth

parity to end of the trajectory variability of litter size increases. Phenotypic averages for NBA by

parities (Figure 18) show similar trend as predicted breeding value curves for NBA (Figure 19). For

example, service sire marked number one has a predicted breeding value curve that increases for a

larger part of trajectory and is positive along parity. Phenotypic average for service sire number one

shows a gradual increase in the number of piglets born alive up to the fifth parity and then a very slow

decrease in litter size appears up to the ninth parity. Therefore, the selection of boars with this type

of breeding value curve could be useful for the improvement of persistency of litter size, although in

the first few parities litter size did not achieve maximum values. On the other hand, the selection of

service sires number 2, 4 or 6 increased litter size in parities two, three and four, and after that litter

size decreased faster compared to service sire number one.

Deviations from the phenotypic average of NBA (Figure 19) showed similar trend as seen on the

previous two figures. For example, deviations from the average NBA illustrate that the curve for

service sire number one increases gradually from the fourthto the tenth parity. Therefore, deviations

from the average NBA for sires confirm and explain breeding value curves for sires.


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Figure 19: Deviations from the phenotypic average for the number of piglets born alive over paritiesfor seven siresSlika 19: Odstopanja od fenotipskega povprecja za število živorojenih pujskov po zaporednih prasit-vah pri sedmih merjascih

One practical question is: How select on litter size ? Besideselection on overall level of litter size,

random regression model enables selection for litter size at specific parity along trajectory. Eigenvalue

analysis shows that the constant term accounted between 85 and 90 % of variability. Thus, between

10 and 15 % variability can be explained with individual litter size curve and that selection on the

shape of production curve would be possible.


5 DISCUSSION

The purpose of this dissertation was to study usefulness of random regression models for the esti-

mation of covariance functions in litter size for pigs. Usually, two approaches were used for the

estimation of genetic parameters for litter size. When a phenotypical expressions are repeatedly mea-

sured over a certain time frame, they can model as a repeated measurements of genetically the same

trait. But often it is more accurate to consider these as the expressions of genetically different, but

correlated traits. The use of a multivariate model to evaluate litter size in pigs has one important

implication. If the hypothesis that genetic correlation for NBA between parities could be different

from unity, different economic weights for different traits have to be defined. Multiple-trait analysis

is not always successful, especially when higher number of parameters are to be estimated. Litter

size as a trait measured more than once in a lifetime of the sowcould be described as a longitudinal

trait. Random regression model is preferred for longitudinal traits. Estimates of genetic parameters

with random regression model were compared with estimates obtained with repeatability model and

multiple-trait analysis.

Large data sets on fertility were used in the analyses. Two data sets were used per farm. In order

to check results, data from three large farms in Slovenia were used. The first data set included litter

records from the first to the sixth parity. The second data setwas extended to the tenth parity. The

second data set was created with the purpose to verify the possibility for the selection on persistency

for litter size. Beside the level of production i.e. litter size, another selection goal could be to maintain

litter size on a high level as long as possible. Because the overall curve for litter size increased up to

the fourth or fifth parity, litter data must be increased to higher parities for this purpose.

5.1 CHOICE OF THE MODEL AND ESTIMATION OF FIXED EFFECTS

Choice of the model and effects included in the model for the number of piglets born alive was based

on significance of the effects, coefficient of determination, number of degrees of freedom for the

model, as well as simplicity of interpretation of effects. Although, modeling of some effects resulted

in relatively small increase of coefficient of determination, we tried to describe them with respect

to their specific relationship to litter size. The percentage of total variability explained with fixed

effects ranged between 9.5 and 10.4 %. It is relatively smallin relation to other important traits in pig

production (daily gain, meat percentage), but within the values found in other studies. Determination

coefficient obtained in our study was much larger in comparison to 1-3 % obtained in the study by

Hermesch et al. (2000). Their fixed part of the model was different. Fixed effects in their model for

NBA included farrowing season as a three month period withina year, sow genotype, insemination

type (natural service or artificial insemination), farrowing unit and age at farrowing fitted as linear

covariable. On the other hand, Duc et al. (1998) reported 32 %of total variability explained with

fixed part of the model that included herd*year*season interaction, parity and service sire breed.

Age at farrowing nested within parity explained the largestpart of variability from all fixed effects.

A need to estimate the effect of age at farrowing within parity was mentioned by Clark and Leman


(1986a). They observed that factors influencing litter sizedid not express their effects similarly in all

parities, and thus they should be investigated within parity to avoid the confounding effects of parity

on these factors. Similar approach with regard to definitionof class levels for parity effect was also

applied by Noguera et al. (1998). They considered three classes of parity effect: the first, the second

and the third (grouping the third to fifth parity). Marois et al. (2000) fitted age at farrowing as a linear

covariate within each parity (six individual class effectsfrom the first to the sixth parity). In the study

by Hermesch et al. (2000) the effect of age at farrowing was also presented as linear covariable, but not

nested within parity. Data set in their study included only the first three parities. Quadratic regression

nested within parity class sufficiently described the effect of age at farrowing on the number of piglets

born alive, noticed by Marois et al. (2000) and Tribout et al.(1998). Graphical description of age at

farrowing and parity on litter size presents a different genetic background for litter size in the first

and the second parity in comparison to higher parities. Thisevidence could be supported with lower

genetic correlations between the first and higher parities,as reported by Johansson (1981), Irgang

et al. (1994) and Duc et al. (1998).

Service sire was modelled as a class effect. Its inclusion tothe model resulted in a substantial increase

of coefficient of determination for the model. Estimates of service sire effect for the number of piglets

born alive showed considerable variability between animals. The estimated difference between the

best and the worst sire ranged between 4.98 and 6.29 livebornpiglets. Similar difference was reported

by Logar (2000). Substantial differences between individual service sires in regard to reproductive

performance were reported by Buchanan and Johnson (1984) and Logar (2000). Therefore a possibil-

ity exists, by culling the less prolific service sires, at least to maintain litter size at a satisfactory level.

More attention should be given to service sires with extremenegative and positive values. Reduced

fertility of some service sires could be a consequence of chromosomal aberration (Locniškar, 1974).

Despite a relatively large number of degrees of freedom for models obtained, due to substantial dif-

ferences among service sires modelling of service sire as fixed effect gave very precise estimate. But

from practical point of view, in future prediction of breeding values service sire could be included as

random effect.

Weaning to conception interval has an obvious influence on the subsequent litter size (Fahmy et al.,

1979; Dewey et al., 1994; Le Cozler et al., 1997). Fahmy et al.(1979) stated that there is a progressive

increase in litter size with the delay of oestrus (after 7 days). Dewey et al. (1994) found a U-shaped

curve with litter size at a minimum for sows conceiving at 7 to10 days after weaning. Le Cozler

et al. (1997) reported that significant differences in the number of piglets born total were observed

according to WCI (P<0.05), with a very marked decrease in litter size when WCI increased from

the fourth to ninth days. Some differences between studies could be the consequence of a different

detection of onset of oestrus, as well as different numbering of days. One reason for litter size decrease

could be late insemination of sows (Rozeboom et al., 1997). Kemp and Soede (1996) showed that

sows which entered oestrus after the fifth day had significantly shorter interval from the onset of

oestrus to ovulation and if the first signs of oestrus were notdetected on time, we can not expect

optimal litter size. The second reason may be simply physiological where some sows had smaller


litters. Sows with the prolonged WCI may probably be in a poornutritional and physiological state,

especially in the first parity sows (Love, 1979). Because of this poor state, they may have smaller

litter size than sows conceiving earlier (due to a smaller number of embryos produced and/or to a

higher embryonic mortality). At still longer WCI, larger liter size could be explained by conception

at the second oestrus. Dewey et al. (1994) reported smaller litter size up to one piglet, when weaning

to conception interval ranged between seven and ten days. LeCozler et al. (1997) described a drop in

litter size after the fourth day after weaning and lasted up to the tenth day. Kemp and Soede (1996)

stated that negative influence of weaning to conception interval was the consequence of suboptimal

mating time regarding the time of ovulation, and not becauseof lover fertility of sows. Determination

of the optimal mating time in sows is difficult. It depends on the detection of oestrus - efficacy and

frequency, onset and length of ovulation and fertilizationabilities of sperm and eggs. Optimal mating

time in relation to litter size is for primiparous sows between zero and 12 hours before ovulation, and

for sows between zero and 24 hours before ovulation. After onset of ovulation fertilization ability

decreases very fast. Onset of ovulation usually occurs between 20 and 60 hours after the start of

oestrus, where this interval decreases in sows that conceived later after weaning. Weitze et al. (1994)

found significant difference of interval from onset of oestrus to ovulation in sows that showed oestrus

signs before and after the fourth day after weaning. Although some authors (Babot et al., 1994; Logar

et al., 1999) assumed that the effect of previous weaning to conception interval on litter size is linear,

a recent study from Marois et al. (2000) showed that this effect is curvilinear with the lowest litter

size for previous WCI interval of 7 to 10 days and cannot be accommodated by linear, quadratic

regression or other common regressions. The best solution for this effect may be in finding some

function showing specific pattern of this effect. This function should probably combine few functions

like cosines, square root, exponential function, and others. It is desirable that this combined functions

should be linear.

Mating season defined as year month interaction enables the monitoring of litter size from month to

month. It allowed a relatively fast reaction in situations where some effects included in the season

effect caused litter size decrease. Beside the climatic components (temperature, photoperiod, humid-

ity), season effect includes other unknown sources of variation. Non periodically changes of litter

size by season suggest that other environmental factors, like management and nutrition, could be the

reason for those changes. This finding was in line with Malovrh et al. (1996), where mating season

was defined as year month interaction too.

The effect of previous lactation length on subsequent litter size was presented with linear regression.

Estimated linear regression coefficients for the previous lactation length ranged between 0.026 and

0.041. Similar results were obtained by Xue et al. (1993), Logar et al. (1999) and Marois et al. (2000).

Linear regression sufficiently described relation betweenthe previous lactation length and litter size,

except for the interval up to 18th day of lactation. This findings are in line with the study from Kovac

et al. (1983) that showed more than twice higher regression coefficients if records with short lactation

length (under 18 days) were excluded from the analysis. Logar et al. (1999) noticed that records with


short lactation length did not give the best fit of regressionline to the records with longer lactation.

They suggested that litters in which lactation was short should not be included in genetic evaluation.

Sow genotype effect presents the effect of purebred and crossbred sows. As expected, crossbred sows

had larger litter than purebred ones; between 0.36 and 0.76 piglets per litter more than mean of both

purebred genotypes. Therefore, between 4 and 7 % of heterozis in relation to mean of purebreds

was observed. The obtained heterozis is in agreement with finding of Gordon (1997), who reported

heterozis for litter size in magnitude between 5 and 25 %, depending on genetic difference among

breeds used in crossbreeding scheme. Larger litter size obtained in crossbred sows was in agreement

with previous studies from Logar et al. (1999) and Logar and Kovac (2001b).

Random part of the model included direct additive genetic, permanent environmental and common

litter environmental effect. Small estimate of maternal genetic effect was the reason for discarding

them from the model. Roehe and Kennedy (1993a) accented thatmaternal genetic effects can have a

high influence on genetic improvement of litter size, even when maternal heritability is low compared

to direct heritability, depending on the genetic correlation between maternal and direct genetic effects.

Correlation estimates between the two genetic effects werenegative and varied from - 0.23 to - 0.87 in

a previous study at another farm in Slovenia (Kovac and Sadek-Pucnik, 1997). Therefore, on the base

of this previous study maternal genetic effect was estimated. Here, estimates of maternal additive

genetic effect were negligible (ranging between 0.0002 and0.0047). These estimates were much

smaller than values reported in other studies (Southwood and Kennedy, 1990; Ferraz and Johnson,

1993). At the same time, genetic correlations were small andmainly positive or close to zero. Small

negative genetic correlation between direct and maternal genetic effect was found only on farm C (-

0.10) and this value was similar to the value - 0.03 obtained by Perez-Enciso and Gianola (1992) in

one strain of Iberian pig. Therefore, we assumed that maternal genetic effect was not significant in

any data set analysed and it was excluded from further analyses. The estimated genetic correlations

between direct additive genetic and maternal genetic effects for NBA in the study from Chen et al.

(2003) ranged between - 0.27 and - 0.70. Despite these relatively high negative genetic correlations

Chen et al. (2003) concluded that due to little change in ranking of sows on estimated breeding

values, models with only direct genetic effects are sufficient. The small maternal genetic effects may

be explained by the large amount of crossfostering practiced, which means that litter mates at birth

do not share the same postnatal environment (Gu et al., 1989;Crump et al., 1997).

5.2 COMPUTATION REQUIREMENTS

Computation demands represent one of the key factors for thechoice of methods for the estimation

of covariance components, especially in the large data sets. Repeatability model needs in general

at least the time and computing resources for the estimationof genetic parameters. Much more

computing time is required for multiple-trait analysis. Due to high genetic correlations in higher

parities, multiple-trait analysis was not always successful. A potential problem of all maximization

methods like REML is a convergence to points other than the global maximum using different starting


values for the same data (Groeneveld and Kovac, 1990). Therefore, several runs with different starting

values were needed to obtain the global maximum with the multiple-trait analysis.

Random regression model is cost effective in computing time, as well as more robust than the

multiple-trait analysis. Computation problems in multiple-trait analysis due to high correlations be-

tween litter size in higher parities were avoided in random regression model. Therefore, inclusion of

higher parities in data set was possible in RRM approach.

5.3 REPEATABILITY MODEL

Repeatability model was used as a frequent procedure for theestimates of genetic parameters for litter

size. Estimates of phenotypic variances ranging between 6.64 and 8.07 were in agreement with liter-

ature estimates (Ferraz and Johnson, 1993; Alfonso et al., 1997; Crump et al., 1997; Hanenberg et al.,

2001). Lower estimates of phenotypic variance for the number of piglets born alive (5.05 - 6.31) were

reported by Chen et al. (2003). Pig losses accounted during the first two days reduced the phenotypic

variance for NBA (Kaplon et al., 1991), what could be one of the explanations for a difference in

variance estimates between farms. Hermesch et al. (2000) found phenotypic variance in the first three

parities in the range from 5.90 to 6.11 using recording procedure where piglets were counted three

days after farrowing. Therefore, recording procedure affects estimates of phenotypic variance for

NBA. Similar as for phenotypic variance, lower estimates ofdirect additive genetic variance than in

our study, ranged between 0.40 and 0.59, were reported by Chen et al. (2003). The estimates of direct

heritability obtained by a repeatability model ranged by farms between 0.09 and 0.11 and were in

agreement with the average of many literature estimates reviewed by Rothschild and Bidanel (1998).

Lower heritability estimates regarding this study were also obtained in the study from Alfonso et al.

(1997), and ranged between 0.05 and 0.07. Although heritability estimate is well known to be low,

we can stress that there is no effect which explains similar part of variability as direct additive genetic

effect, and in case of litter size the 10 % of explained variability is not so low.

Common litter environmental effect had a relatively low estimate, ranging as ratio, between 0.01

and 0.02. Similar estimates were also found by Mercer and Crump (1990) and Crump et al. (1997).

Small estimates were often explained as a consequence of crossfostering immediately after farrowing

(Crump et al., 1997). Including common litter environmental effect as random effect is not so often

the case, probably due to confounding with maternal effect.

Estimates of permanent environmental variance were in linewith the study from Chen et al. (2003)

which reported values between 0.34 and 0.49. Permanent environmental effect presented as ratio

in phenotypic variance agrees with most studies (Andersen,1998; Götz, 1998; Tribout et al., 1998),

where it ranged between 5 and 8 %. In contrast, these estimates were considerably smaller than those

in the study from Ferraz and Johnson (1993) which found that permanent environmental effect of

sows explained 16 to 17 % of total variability.


5.4 MULTIPLE-TRAIT ANALYSIS


Multiple-trait analysis as the alternative to repeatability model was conducted for later compari-

son with estimates obtained with random regression model. Covariance components obtained with

multiple-trait analysis were generally in agreement with estimates from the literature (Roehe and

Kennedy, 1995; Rothschild and Bidanel, 1998). Similar findings of heritability decline in the second

parity were found by Hanenberg et al. (2001). Generally, estimates of variances with multiple-trait

model were slightlyhigher than those obtained with repeatability model, and this suggests that the

model become better. However, this result is not in agreement with some previous studies where heri-

tabilities obtained with multivariate model were lower than those obtained by univariate repeatability

model (Alfonso et al., 1997; Hanenberg et al., 2001).

5.4.2 Correlations

Lower genetic correlations between the first and the second or the third parity (0.55 - 0.74) compared

to correlations obtained in our study were presented by Alfonso et al. (1994) and Hermesch et al.

(2000). Irgang et al. (1994) found even lower genetic correlations between the first and the second

parity (0.50). Duc et al. (1998) stated that moderate genetic correlations between the first and the

later parities indicate a slightly different genetic control in the first parity. However, Hermesch et al.

(2000) reported high genetic correlation (0.95) between NBA in the second and the third parity and

suggested these traits (parities two and higher) could be treated as repeated measurements.

Direct additive genetic correlations obtained in this study are in agreement with the results from

Hanenberg et al. (2001). They reported an increase from 0.79between the first and the second parity

to 0.96 between the fifth and the sixth parity.

Estimates of phenotypic correlations were comparable withthose indicated by Alfonso et al. (1994)

and Duc et al. (1998). Phenotypic correlation for the numberof piglets born alive between litter size

in the first and higher parities of 0.18 was obtained in the study of Logar and Kovac (2001a).

5.5 RANDOM REGRESSION MODEL

Litter size in pigs, as a trait measured more than once duringthe sow’s lifetime, could be considered

as a longitudinal trait. The random regression i.e. covariance function approach gives the opportu-

nity to fit more complete models which also accommodate time dependent changes in genetic and

environmental effects. Random regression is an accurate and robust technique for the estimation of

genetic parameters even when data is not evenly distributedacross parities, as often happens in large

field data sets. This provides a perspective for random regression of genetic covariance function using

data recording from extensive industries.


Legendre polynomials of different order were used in randomregression model in order to fit variabil-

ity in the population along the litter size trajectory. Orthogonal Legendre polynomials do not require

prior assumptions about the shape of trajectory. They are flexible enough and as such appropriate for

the analysis. Cubic Legendre polynomials with four terms were sufficient enough to explain almost

all variability in the population.

5.5.1 Eigenvalue analysis

As pointed out above, two data sets were compared. Eigenvalues analysis showed that the inclusion

of higher parities to the analysis led to the considerable increase of a part of variability explained with

the production curve of individual animal. Sufficiently large genetic variability for individual shape

of the production curve indicates that random regression model could be used for the selection on the

level (litter size) and the shape of production curve for litter size.

Eigenvalues of genetic covariance function showed that theconstant term accounted for between 85

and 90 % of total genetic variability for the number of piglets born alive. This means that most of

genetic differences between animals were a consequence of differences in the overall level of litter

size. Between 10 and 15 % of variability in our study was covered by linear, quadratic and higher

coefficients. Therefore, animals differ genetically regarding the shape of their litter size curve.

The obtained 10 to 15 % of variability explained with the individual production curve for litter size of

animals for data set which included ten parities is comparable with the results from Malovrh (2003).

She reported that the shape of growth curve causes 10 % of the genetic variability in boars and

more than 15 % in bulls. Further, the author suggested that the obtained percentages of the genetic

variability showed the possibility for utilisation in the selection on growth shape. Therefore, 10 to

15 % of genetic variability for the number of piglets born alive could be used for changing the shape

of litter size curve in pigs. Genetic eigenvalues showed that altering the shape of litter size curve

would be more difficult than changing the overall level for litter size.

Eigenvalues of covariance functions showed the cubic Legendre polynomial with four regression co-

efficients to be flexible enough to cover 99 % additive geneticeffect. For environmental effects even

the quadratic Legendre polynomial with three coefficients could be enough. Lower order of polyno-

mials considerably decrease computational demand. Kirkpatrick et al. (1990) pointed out that even

lower order of orthogonal polynomials usually describe variability sufficiently. Therefore, describing

the some effects with fewer random regression coefficients would be preferable because of simplicity

(parsimony) and lower computational requirements.


Random regressions on parity were included for direct additive genetic, permanent environmental

and the common litter environmental effect. Estimates of genetic and environmental variances from


random regression model with cubic Legendre polynomial were in agreement with multiple-trait anal-

ysis, although some slightly higher estimates were observed. Genetic and environmental variances

after the eight parity should be considered with caution. The number of records after the eight parity

was small, and they probably did not contain enough information for the proper estimation of covari-

ance functions. Random regression is probably less stable at the edges of the trajectory (van der Werf

et al., 1998), possibly due to a reduced amount of data in these areas. In comparison to multiple-trait

model, random regression model is more correct from statistical point of view, because it includes per-

manent environmental effect of a sow, which contributes substantially in the proportion of explained

phenotypic variability. On the other side, although repeatability model includes permanent environ-

ment in the model, it does not allow any change of correlations over parities. Comparing all three

models, there is a trend of increased proportion of explained variability from repeatability through

multiple-trait model to random regression model. Heritabilities also increased with the complexity of

the model.

5.5.3 Correlations

Estimates of correlations in random regression model were comparable with those obtained with

multiple-trait model. Permanent environmental correlations can be estimated in random regression

model. Correlation estimates for different random effectsshowed a pattern of decrease of correla-

tions if the distance between parities increased. Correlations between adjacent parities for random

effects were higher than correlations between two distant parities. Estimates of direct additive genetic

correlations obtained with random regression model were slightly higher than those obtained with

multiple-trait model. In general, direct additive geneticcorrelations ranged from 0.40 to 0.90. The

structure of genetic correlations showed that litter size in different parities is not the same trait genet-

ically. Additionally, genetic correlations confirmed the existence of individual genetic variability, as

well as adequacy of the use of random regression for genetic evaluation.


Reliable estimates of variance components are of great importance in any livestock improvement

scheme. Estimates of heritability and other random effectsare a function of variance components

and are, in general specific for a particular population and period of time. Estimates from large field

data sets are not always in agreement with those from designed research or controlled test stations.

Random regression model offers new criteria for the selection on litter size. Estimates of breeding

values by parities could be combined into total breeding value including economic weights of litter

size for individual parity. Additionally, using random regression coefficients which describe specific

feature of animal production curves, RRM can provide selection on the level, as well as on the per-

sistency of litter size. Additional studies need to be conducted to analyze the relative improvement in

the prediction of breeding values using random regression model.


6 CONCLUSIONS

The following conclusions can be drawn from the study:

- Fixed part of the model for the number of piglets born alive explained around 10 % of total variabil-

ity.

- The number of piglets born alive by the random regression approach had low heritability. Heritabil-

ity estimates obtained with random regression analysis were slightly higher in comparison to those

obtained in repeatability and multiple-trait model. Slow increasing trend of heritability estimates from

simple repeatability model to random regression model suggests the improvement of modelling.

- Covariance components for the number of piglets born aliveobtained by random regression ap-

proach have changed over parities and are in agreement with multiple-trait analysis. Direct additive

genetic correlations were similar for both approaches, andshowed that litter size in different parities

could be genetically a different trait.

- Random regression model is preferred to multiple-trait model due to its robustness and lower com-

putational demand. Due to high genetic correlations between later parities, analysis of data set which

included higher parities is only possible with random regression model.

- The proportion of variation explained with the individualgenetic curve of sows increased with

prolongation of the trajectory (number of parities). The existing 10 to 15 % of genetic variation for

the shape of the production curve indicates that random regression model could be used for selection

on the level and shape of production curves for litter size.

- Eigenvalue analysis showed cubic Legendre polynomial with four regression coefficients to be flex-

ible enough to cover 99 % of additive genetic and common litter environmental variability. For

permanent environmental effect quadratic Legendre polynomial could be sufficient.

- The predicted breeding values from random regression coefficients for direct additive genetic effect

allowed selection on the overall level of production, on litter size at specific parity, and on the shape of

production curve or the persistency of litter size. Geneticeigenvalues showed that altering the shape

of litter size curve would be more difficult than changing theoverall level of litter size.

- Further research should be directed toward practical application of random regression model in

genetic evaluation for litter size, firstly to use service sires with higher estimates of breeding values

for the number of piglets born alive by parities.


7 SUMMARY (POVZETEK)

7.1 SUMMARY

Litter size is one of the most important traits in overall reproductive efficiency of breeding sows.

The triat has low heritability, is measured only in females and is expressed relatively late. Genetic

improvement for litter size is possible due to large enough direct additive genetic variance, the avail-

ability of records on many relatives and short generation interval. Litter size is affected by many

environmental and genetic factors and interaction betweenthem.

Improvement of litter size is mainly the consequence of application of mixed model methodology.

Genetic evaluation of litter size is generally based on repeatability model because of simplicity. Re-

peatability model assumes complete genetic correlations between litter size in different parities and

a constant variance along the trajectory. When genetic correlations are substantially lower than one,

multiple-trait model is preferred. However, multiple-trait analysis is not always successful, especially

when larger number of parameters have to be estimated. In data sets with higher parities, there are

high genetic correlations between litter sizes in later parities. Further, by defining litter size in succes-

sive parities as a different trait we must find the answer to the following question: How to combine

these traits into breeding objective ? Litter size is a traitusually measured more than once in sow

lifetime, and therefore could be considered as longitudinal trait. More appropriate analysis for a lon-

gitudinal trait is to fit a set of random regression coefficients describing production over time for each

animal, resulting in random regression model. Such model has not been studied for litter size yet, but

had been used for feed intake and growth in pigs.

The goal of this study was to determine covariance functionsfor litter size i.e. number of piglets born

alive using random regression model. Further, the aim was tocompare it with estimates of genetic

components obtained by using repeatability and multiple-trait model.

Data included litter records between January 1990 and December 2002 for two farms (B and C), and

from January 1992 to December 2002 for one farm (A). Two data sets were used per each farm. The

first data set contained litter records from the first to the sixth parity, while the second was extended

to the tenth parity. Only the first data set was prepared for multiple-trait analysis, and the second to

study features of litter size in the higher parities. Individual records were excluded from the analysis

if explanatory variables were outside expected interval. Thus the lactation length was limited to 60

days. The weaning to conception interval longer than 80 dayswas excluded. After editing, smaller

data set had 61415 records on farm A, 99512 on farm B and 85986 records on farm C. Extended data

set included 70033 records for farm A, 118079 records for farm B, and 99411 records for farm C.

Service sires with less than 10 litters were grouped by genotype. Data structure by farm was similar.

Farms differed in the number of litters per sow, as expected.The number of sows selected from

same litter was similar and ranged between 1.40 and 1.50 between farms. Four sow genotypes were

included: Swedish Landrace, Large White and both crossbredlines between the two breeds.

Repeatability model served for the determination of fixed effects of the models. Fixed part of the


model was developed using GLM procedure from SAS statistical package. In the process of devel-

opment of the model several alternative models were fitted. The effects were fitted as class effects,

linear regression or quadratic regression. The model was chosen on the base of coefficient of determi-

nation, degrees of freedom, significance and proportion of variation explained by the studied effect.

Fixed effects with classes included in the models for genetic evaluation of the number of piglets born

alive were: service sire, weaning to conception interval, mating season, and sow genotype. Age at

farrowing was modelled as a quadratic regression nested within parity. The previous lactation length

was fitted as a linear regression. The weaning to conception interval and previous lactation length

were included in the model for higher parities.

The random part of the model consisted of common litter environmental, permanent environmental

and direct additive genetic effect. The study of maternal effects showed a negligible estimate for

the maternal genetic effects. Therefore, it was not included in the model. Equivalent multiple-trait

model included same fixed effects as repeatability with the exception of parity class. Random effects

mentioned above in repeatability model were also included in multiple-trait model, except permanent

environmental effect. Estimates of genetic parameters obtained with repeatability and multiple-trait

model were used for later comparison with random regressionmodel estimates. Fixed part of the

random regression model included same effects as repeatability model. Direct additive genetic effect,

common litter environmental and permanent environmental effects were fitted as random regression

on parity, using Legendre polynomials. Linear to cubic Legendre polynomials were fitted. Estima-

tion of (co)variance components for different models was based on the residual maximum likelihood

(REML) method, using VCE-5 software package. The analytical gradients of the likelihood func-

tion are explicitly calculated in optimization procedure for maximizing likelihood. Eigenfunctions

and corresponding eigenvectors were calculated from covariance matrices. Eigenvalues of genetic

and environmental covariance matrices quantify the relative importance of each order of Legendre

polynomials. Predicted random regression coefficients fordirect additive genetic effect served for the

prediction of breeding values of animals along the trajectory.

Fixed part of the model explained between 9.5 and 10.4 % totalvariability. All included effects were

significant (P<0.0001), and the importance of each individual effects as a deviation in coefficient of

determination from the coefficient of determination obtained with full model showed that two most

important effects for litter size were age at farrowing nested within parity and service sire. Two effects

with intermediate influence on litter size were weaning to conception interval and mating season. At

the end, two effects with the smallest influence on litter size were previous lactation length and sow

genotype.

Similarly, litter size was influenced by age at farrowing andparity. The phenotypic values showed

that litter size depends on age at farrowing. Litter size in the first and in the second parity expressed

specific pattern. Wide range of possible sow farrowing ages within parity was the reason to combine

those two effects. In the first parity litter size increase upto 400 days, while in the second parity

it was up to the age of about 550 days. After the second parity the effect of age on litter size was

less evident and therefore parities from the third to the tenth were considered as one class. Quadratic


regression within parity class sufficiently described the effect of age at farrowing. Estimated linear

regression coefficients for age at farrowing nested within parity class ranged between 0.12 and 0.19

for parity class one (the first parity); between 0.03 and 0.09for parity class two (the second parity),

and between 0.002 and 0.003 for parity class three (parity three and higher). Estimated quadratic

regression coefficients were negative and much smaller thanlinear regression coefficients.

Service sire effect on the number of piglets born alive showed considerable variability. The estimated

differences between the highest estimate (the best servicesire) and the lowest estimate (the worst ser-

vice sire) ranged between five and six piglets born alive. Most of the estimates for service sire (90 %)

ranged from -1 to +1. More attention should be payed to service sires with extreme values. Therefore

a possibility exists, by culling less prolific boars, to at least maintain litter size at a satisfactory level.

Relationship between weaning to conception interval and the number of piglets born alive showed

specific decrease in litter size between the day 6 and the day 7after weaning. In relation to the fifth

day of weaning to conception interval, if sows conceived between the day 6 and the day 9, litter size

decreased up to -0.75 piglets born alive. This decrease could be important from economical point

of view, because it included from 10 to 20 % of all litters. Thedecrease could be explained partly

with suboptimal mating time, when sows were inseminated toolate. Secondly, one part of the sows

conceived later due to physiological reasons. Therefore, in sows that did not conceive within five

days after weaning, the oestrus should be checked at least twice daily.

The mating season effect on litter size usually explains twosources of variation. Firstly, there are

long-term changes as a consequence of improved environment, management and selection. Secondly,

litter size can oscillate due to short-term changes usuallyrelated with climate, as well as changes in

technology practices and other unknown sources of variation. The studied farms were located in the

same climate region, therefore differences in the number ofpiglets born alive between farms could

probably be referred to differences in management practices (oestrus synchronization with service

sire).

The effect of previous lactation length on the number of piglets born alive was presented with linear

regression which was favourable between 18 and 30 days, where the majority of the data exists.

Outside this interval linear regression did not describe litter size well. Linear regression coefficients

for lactation length ranged between 0.026 and 0.041. Although linear regression is not the best choice

for the presentation of this effect on the whole interval (especially under 18 days), preliminary analysis

showed that nesting of linear regression of previous lactation length within lactation intervals did

not change ranking of animals. Therefore, linear regression along the whole interval seemed to be

sufficient.

Sow genotype estimates were compared in relation to genotype Swedish Landrace. Crossbred sows

had larger litters than purebreds. Between 0.36 and 0.76 liveborn piglets per litter more than mean

of both purebred genotypes, which means between 3 and 8 % of heterozis in relation to mean of

purebreds. From all observed effects, effect of sow genotype had smaller influence on the litter size in

this study. This is probably due to small differences in reproductive performance between genotypes

used in analyses.


Genetic parameters using repeatability model were estimated under assumption that genetic corre-

lations between parities were one. The estimates of heritability ranged between 9.4 and 10.6 %.

Common litter environmental effect as ratio explained up to2 % of variation. Small magnitude could

be a consequence of small full-sib groups. Estimates of permanent environmental effect was higher

and accounted between 5 and 6 % of the phenotypic variance.

Multiple-trait analysis was performed only for the smallerdata set that included records from the first

to the sixth parity. Direct additive genetic correlations between different parities showed that litter size

in the first parity could be genetically different trait thanlitter size in higher parities. Direct additive

genetic correlations between litter size in the first and later parities ranged between 0.85 and 0.53.

Genetic correlations were the highest between adjacent parities (0.80 - 0.99) and decreased as the

interval between parities increased. Estimates of phenotype correlations were much lower, varying

from 0.11 to 0.22. Heritability estimates for the number of piglets born alive were as expected.

They changed slightly by parities and ranged between 0.10 and 0.14. Estimates increased from the

first to fifth parity. In general, heritability estimates obtained with multiple-trait model were slightly

higher than those found in univariate repeatability model.The ratio of the common litter environment

variance with respect to the total variance varied between 0.01 and 0.06.

Random regression model was performed for the smaller and larger data set. Random regression

models with different order of Legendre polynomial were compared and selected on the basis of

eigenvalue analysis. Eigenvalues for Legendre polynomials from linear to cubic power for direct

additive genetic, permanent environmental and common litter environmental effect were computed.

The eigenvalues of covariance matrices of random effects quantifyed the relative importance of each

order of Legendre polynomials. The eigenvalues showed thatquadratic Legendre polynomials with

three regression coefficients could be sufficient. The eigenvalues of genetic covariance functions for

smaller data set showed that the constant term accounted between 90 and 95 %. Extending data to the

tenth parity caused a decrease in explained proportion of variance by the zero-th eigenvalue for direct

additive genetic effect. The constant term explained between 85 and 90 % of variability. The rest

(between 10 and 15 %) of variability was explained by individual production curves of sows. This

percentage of genetic variability could be interesting forselection on the shape of production curve

for litter size. Random regression methodology is of higherimportance if more litters are included in

the data set because litter size decreased after the 6th parity and the persistency of litter size could be

studied.

Estimates of genetic and environmental variances, as well as ratio in phenotype variance from random

regression model for the small data set were in agreement with estimates from multivariate analysis.

Results at the end of the trajectory in the extended data set should be looked with caution. Random

regression is probably less stable at the edges of the trajectory, possibly due to a reduced amount of

data. Thus, including higher parities in data set estimatesof genetic parameters in the smaller data

set can be confirmed and used with more reliability. Heritability estimates for the larger data set

increased a little more. It suggests the improvement of the model.

Solutions for the random regression coefficients of the additive genetic effect used to compute the


predicted breeding values for the number of piglets born alive by parities and thus produce genetic

merit functions. Genetic merit functions differed in the level, as well as in the shape. Breeding values

using random regression model can be estimated at differentages (parities). Therefore, with random

regression model litter size could be selected differently. It could be selected on the overall level

of production along all productive intervals, or on production level at specific parity. The existence

between 10 and 15 % of genetic variation for the shape of the production curve indicates that random

regression model could also be used for selection on the shape of production curve for litter size.

Covariance components for the number of piglets born alive obtained by random regression approach

changed over parities and were in agreement with a multiple-trait analysis. The proportion of variabil-

ity explained with the individual genetic curve of sows increased with prolongation of the trajectory

(number of parities). A random regression model is preferedto multiple-trait model due to its robust-

ness and lower computational demand. Estimation of geneticparameters by random regression model

was not so time consuming as the multiple-trait analysis of litter size. Random regression model was

more robust, since several runs with different starting values were needed for multiple-trait analysis

to obtain the global maximum.


7.2 POVZETEK

Velikost gnezda je ena izmed najpomembnejših lastnosti priplodnosti svinj. Lastnost ima nizko he-

ritabiliteto in se izraža relativno pozno v življenju. Genetski napredek pri velikosti gnezda je možen

zaradi zadostno velike aditivne genetske variance ter kratkega generacijskega intervala. Z uvedbo

metode mešanega modela se je povecala tudi zanesljivost napovedi, ker lahko vkljucujemo podatke

iz vec zaporednih prasitev, podatke sorodnikov in informacijo osorodstvu. V modelu pa lahko od-

stranimo tudi vpliv skupnega okolja pri vzreji plemenskegapodmladka. Na velikost gnezda vplivajo

številni okoljski in genetski dejavniki ter interakcije med njimi. Pri modeliranju lahko upoštevamo le

tiste, ki jih pri spremljanju proizvodnje v rejah beležijo.

Genetsko vrednotenje velikosti gnezda v praksi temelji na ponovljivostnem modelu, predvsem zaradi

enostavnosti. Ponovljivostni model predpostavlja popolne genetske korelacije med velikostjo gnezda

ter identicno varianco pri razlicnih zaporednih prasitvah. Ko so genetske korelacije manjše od ena,

je primernejši veclastnostni model, ko obravnavamo vsako zaporedno gnezdo kot drugo lastnost.

Toda veclastnostna analiza ni vedno uspešna zaradi visokih korelacij pri višjih zaporednih prasitvah.

Problemi se še stopnujejo takrat, ko vkljucujemo višje zaporedne prasitve in je tako potrebno oceniti

veliko število parametrov disperzije. Velikost gnezda je lastnost, ki jo v življenju svinje lahko veckrat

zabeležimo, zato jo lahko oznacimo kot longitudinalno lastnost. Primernejša analiza za longitudinalne

lastnosti je uporaba modelov z nakljucno regresijo, ki opisujejo prirejo živali s proizvodno funkcijo.

Tak model za velikost gnezda še ni bil preucevan, uporabili pa so ga za zauživanje krme, rast živali in

spreminjanje konformacije telesa ter laktacijske krivulje pri lastnostih mleka.

Namen doktorske disertacije je bil razviti statisticni model za število živorojenih pujskov kot mero

velikost gnezda pri prašicih z vkljucitvijo nakljucne regresije, ki omogoca izvrednotenje plemenske

vrednosti tako za nivo kot tudi potek proizvodne funkcije. Model smo primerjali s ponovljivostnim

modelom živali in z veclastnostnim modelom.

Podatki vsebujejo zapise o velikosti gnezda od januarja 1990 do decembra 2002 na farmah B in C ter

od januarja 1992 do decembra 2002 na farmi A. Za vsako farmo smo uporabili dva niza podatkov.

Prvi niz je vseboval zapise o velikosti gnezda od prve do šeste zaporedne prasitve in je služi primerjavi

med veclastnostnim modelom in modelom z nakljucno regresijo. Drugi niz podatkov smo pripravili

kot razširjen prvi niz in je vkljuceval gnezda do vkljucno desete zaporedne prasitve. Velikost gnezda

se povecuje docetrte oziroma pete zaporedne prasitve, kasneje pa pada. Drugi niz podatkov smo

pripravili z namenom preverjanja možnosti selekcije na perzistenco velikosti gnezda. Poleg nivoja

proizvodnje, t.j. velikosti gnezda, je lahko drugi selekcijski cilj cim manjše upadanje velikosti gnezda

po peti zaporedni prasitvi. S tem namenom smo v drugi niz vkljucili tudi podatke o velikosti gnezda

višjih zaporednih prasitev.

Zapise, kjer so pojasnjevalne spremenljivke zavzemale vrednosti izven pricakovanih intervalov, smo

izkljucili iz analize. Dolžino laktacije smo omejili do 60 dni, prav tako smo izkljucili zapise, kjer je

bil poodstavitveni premor daljši od 80 dni. Po urejanju podatkov, je prvi niz podatkov vseboval 61415

zapisov za farmo A, 99512 za farmo B in 85986 za farmo C. Drugi niz podatkov je dodatno zajemal


še gnezda od vkljucno sedme in do desete prasitve ter tako obsegal 70033 zapisov za farmo A, 118079

za farmo B in 99411 za farmo C. Merjasce, ocete gnezd, ki so zaplodili manj kot 10 gnezd, smo po

genotipih združili v skupine. Struktura podatkov je bila pofarmah podobna, število gnezd na svinjo

pa se je med farmami razlikovalo, kar smo tudi pricakovali. Število svinj, odbranih iz istega gnezda, je

bilo podobno in je zavzemalo vrednosti med 1.40 in 1.50. Datoteka s poreklom je bila pripravljena za

tri generacije. Poreklo se je med farmami razlikovalo po številu zapisov, sama struktura pa se glede na

število vkljucenih sorodnikov in delež osnovne populacije ni pomembno razlikovala. Vkljucili smo

svinje štirih genotipov: švedske landrace, large white ternjunih hibridov.

Pri izboru sistematskih vplivov smo se poslužili ponovljivostnega modela živali, saj razlicni poskusi

v nakljucnem delu modela v našem primeru ne vplivajo na sistematski del. Sistematski del modela

smo razvijali s pomocjo procedure GLM v statisticnem paketu SAS. Izbira modela je temeljila na

koeficientu determinacije, stopinjah prostosti, znacilnosti in deležu variabilnosti, ki so jo pojasnili

proucevani vplivi.

V model za genetsko vrednotenje števila živorojenih pujskov smo vkljucili naslednje sistematske

vplive z nivoji: merjasec - oce gnezda, poodstavitveni premor, sezona pripusta in genotip svinje.

Starost ob prasitvi smo modelirali kot kvadratno regresijo, samo ali vgnezdeno znotraj zaporedne

prasitve. Dolžino predhodne laktacije smo opisali z linearno regresijo. Poodstavitveni premor in

dolžino predhodne laktacije sta bila vkljucena v modelu za višje zaporedne prasitve.

Nakljucni del modela je obsegal skupno okolje v gnezdu, permanentno okolje svinje in direktni adi-

tivni genetski vpliv. Primerljiv veclastnostni model je vseboval iste sistematske vplive kot ponovljivo-

stni, z izjemo zaporedne prasitve. Zgoraj omenjeni nakljucni vplivi ponovljivostnega modela so bili

vkljuceni tudi v veclastnostni model, z izjemo vpliva permanentnega okolja svinje. Ocene genetskih

parametrov, dobljene s ponovljivostnim in z veclastnostnim modelom živali so bile uporabljene pri

kasnejših primerjavah z ocenami, dobljene z modelom z nakljucno regresijo. Sistematski del modela

z nakljucno regresijo je vkljuceval enake vplive kot ponovljivostni model.

Pri razvoju nakljucnega dela modela se je izkazalo, da so ocene maternalnega aditivnega genetskega

vpliva zanemarljivo majhne, zato ga v koncni model nismo vkljucili. Ocene maternalnega aditiv-

nega genetskega vpliva, kot deleža fenotipske variance, sobile majhne in so se gibale med 0.02 %

na farmi B in 0.47 % na farmi A. Hkrati so bile nizke tudi korelacije med direktnim in maternalnim

aditivnim genetskim vplivom, ki so bile vecinoma pozitivne tako na farmi A kot na farmi B. Nizke,

vendar negativne, genetske korelacije med direktnim in maternalnim genetskim vplivom so bile tudi

na farmi C (-0.10). Majhen delež in nizke korelacije med aditivnima genetskima vpliva so bili razlog,

da smo matenalni aditivni genetski vpliv v nadaljnjih analizah zanemarili. Tako je v modelu brez ma-

ternalnega genetskega vpliva, vpliv skupnega okolja v gnezdu poleg okoljske vseboval tudi genetsko

komponento.

Direktni aditivni genetski vpliv, vpliv skupnega okolja v gnezdu in permanentnega okolja so bili

opisani z nakljucno regresijo. Neodvisnocasovno spremenljivko v tej analizi predstavlja zaporedna

prasitev. Pri modeliranju proizvodnih funkcij v nakljucnem delu modela smo uporabili Legendrove

polinome od prve do tretje stopnje. Ocene komponent variance za razlicne modele so temeljile na


metodi najvecje zanesljivosti ostankov (REML), z uporabo VCE-5 programskega paketa. Analiticni

gradienti funkcije zanesljivosti so bili izracunani z optimizacijsko proceduro za najvecjo zanesljivost.

Lastne vektorje in pripadajoce lastne vrednosti smo izracunali iz matrik kovarianc nakljucnih regresij-

skih koeficientov. Lastne vrednosti matrik kovarianc za genetske in okoljske vplive merijo relativno

pomembnost vsake stopnje Legendrovih polinomov. Napovedani nakljucni regresijski koeficienti za

direktni aditivni genetski vpliv služijo za napoved plemenske vrednosti živali vzdolž zaporednih pra-

sitev.

S sistematskim delom modela smo pojasnili med 9.5 in 10.4 % celotne variabilnosti. Vsi vkljuceni

vplivi so bili znacilni (P<0.001). Na osnovi parcialnih koeficientov korelacije sklepamo, da sta k

pojasnitvi variabilnosti najvec prispevala starost ob prasitvi vgnezdena znotraj zaporedne prasitve in

merjasec - oce gnezda, nekoliko manj sta prispevala poodstavitveni premor in sezona pripusta, še

manj pa dolžina predhodne laktacije in genotip svinje. Merjasec in sezona pripusta sta vpliva, ki sta

porabila tudi najvec stopinj prostosti. Analize smo opravili na vseh treh farmah in kljub razlicnim

tehnologijam reje prišli do podobnih zakljuckov.

Ocene genotipov svinj smo prikazali kot odstopanje od pasmešvedska landrace. Pasma large white

je bila v primerjavi s švedsko landrace slabša za -0.32 na farmi B do -0.72 živorojenega pujska na

gnezdo na farmi C. Hibridne svinje so imele vecja gnezda v primerjavi s švedsko landrace, boljše

so bile za 0.01 na farmi C do 0.50 na farmi B. Pri primerjavicistopasemskih in hibridnih svinj so

bile slednje pricakovano boljše, in sicer so imele za 0.36 do 0.76 živorojenih pujskov vec na gnezdo

kot znaša ocenjeno povprecje obeh pasem, ki sodelujeta v križanju, kar predstavlja med 3 in 8 %

heterozis.

Vpliv sezone pripusta na velikost gnezda obicajno vkljucuje dva vira variabilnosti. Dolgorocne spre-

membe so bile posledica izboljšanja življenjskega okolja,gospodarjenja, prehrane in selekcije. Ob

vkljucitvi nakljucnih komponent modela, se genetski trend iz sezone odstrani. Velikost gnezda pa se

lahko spreminja tudi zaradi kratkorocnih sprememb, obicajno povezanih s klimatskimi spremembami

in tudi s spremembami v tehnologiji in ostalimi neznanimi viri variabilnosti. Ker se vse farme naha-

jajo v podobnem klimatskem obmocju, so razlike v številu živorojenih pujskov med farmami verjetno

posledica z razlikami v gospodarjenju in tehnologiji.

Vpliv merjasca - oceta gnezda na število živorojenih pujskov je pokazal precejšno razpršenost. Oce-

njene razlike med najboljšim in najslabšim merjascem - ocetom gnezda so se gibale med 4.98 in 6.29

živorojenih pujskov. Vecina ocen (90 %) za merjasce - ocete gnezda je zavzelo vrednosti od -1 do +1.

Bolj pozorni pa moramo biti pri merjascih - ocetih gnezda z ekstremnimi vrednostmi. Obstaja mo-

žnost, da z izlocevanjem manj plodnih merjascev, vsaj vzdržujemo velikostgnezda na zadovoljivem

nivoju.

Povezava med poodstavitvenem premorom in številom živorojenih pujskov izraža specificen padec

velikosti gnezda med 6. in 7. dnevom po odstavitvi.Ce so bile svinje osemenjene med 6. in 9. dnem,

je bil padec nad -0.75 živorojenih pujskov. Ta padec je lahkopomemben z vidika ekonomicnosti, saj

vkljucuje 10 do 20 % vseh gnezd. Padec lahko delno pojasnimo z optimalnim casom pripusta, ko so

svinje osemenjene prepozno.


Zaradi velikega razpona starosti ob prasitvi vgnezdene znotraj zaporedne prasitve, smo združili ta dva

vpliva. V prvi zaporedni prasitvi se velikost gnezda povecuje do 400. dneva, medtem ko se v drugi

do 550. dneva. Prva in druga prasitev kažeta izrazitejšo povezavo med starostjo in velikostjo gnezda.

Po drugi prasitvi je bil vpliv zaporedne prasitve manj opazen, zato smo prasitve od tretje od desete

združili v skupni razred. Vpliv starosti ob prasitvi je zadovoljivo opisala kvadratna regresija znotraj

zaporedne prasitve. Ocene linearnih regresijskih koeficientov za starost ob prasitvi vgnezdene znotraj

zaporedne prasitve, so zavzemale vrednosti med 0.12 in 0.19za prvi nivo zaporedno prasitve (prva

zaporedna prasitev); med 0.03 in 0.09 za drugo nivo zaporedno prasitve (druga zaporedna prasitev),

in med 0.002 in 0.003 za tretji nivo zaporedno prasitve (tretja in višje zaporedno prasitev). Ocenjeni

kvadratni regresijski koeficienti so bili negativni in pricakovano veliko manjši kot linearnicleni.

Vpliv dolžine predhodne laktacije na število živorojenih pujskov smo pojasnili z linearno regresijo.

Linearni regresijski koeficienti so zavzemali vrednosti med 0.026 in 0.041. Linearna regresija dobro

opiše povezavo predvsem na intervalu, kjer je najvec podatkov (nad 18 dnevi). Kljub temu so predho-

dne analize pokazale, da vnezdenje linearne regresije dolžine predhodne laktacije znotraj laktacijskih

intervalov, ni spremenilo rangiranja živali, zato je bila linearna regresija zadovoljiva pri opisu pove-

zave na celotnem intervalu.

V ponovljivostnem modelu pri ocenjevanju komponent kovariance predpostavljamo, da so genetske

korelacije med velikostjo gnezda v razlicnih zaporednih prasitvah enake ena in zaporedne prasitve

obravnavamo kot ponovitve. Tako ocenjene heritabilitete so zavzemale vrednosti med 0.094 in 0.106.

Vpliv skupnega okolja v gnezdu je pojasnil med 0.0 in 2.0 % fenotipske variabilnosti. Tako majhen

delež za ta vpliv je lahko posledica strukture podatkov, sajje bilo iz istega gnezda vzrejeno in je prasilo

v povprecju med 1.40 in 1.50 svinj. Delež variance vpliva permanentnega okolja je predstavljal med

5 in 6 %.

Veclastnostno analizo smo izvedli zaradi primerjave z rezultati iz modela z nakljucno pregresijo. He-

ritabilitete za število živorojenih pujskov so se med zaporednimi prasitvami razlikovale, ocene so

se povecevale od 10 na 14 % do pete prasitve, v šesti pa sledi rahel padec. Ocene heritabilitet z

veclastnostim modelom so bile v primerjavi z ocenami z enolastnostnim ponovljivostnim modelom

nekoliko višje. Ocene direktnih aditivnih genetskih korelacij za velikost gnezda v prvi in kasnejših

prasitvah zajemajo vrednosti med 0.85 za drugo prasitev in 0.53 za šesto prasitev. Genetske korela-

cije so najvecje med sosednjimi zaporednimi prasitvami (0.80 – 1.00) in se zmanjšujejocim bolj so

zaporedne prasitve oddaljene. Razlike pri genetskih korelacijah za velikost gnezda med razlicnimi

zaporednimi prasitvami kažejo na to, da bi velikost gnezda pri mladicah lahko smatrali kot genetsko

drugacno lastnost kot velikost gnezda po drugem gnezdu. V višjih zaporednih prasitvah pa je ge-

netska korelacija vedno višja in so torej gnezda prakticno ponovitev meritve za isto lastnost. Ocene

fenotipskih korelacij so bile bistveno nižje od genetskih,variirale so med 0.11 in 0.22. Tudi pri drugih

komponentah kovariance smo opazili podoben trend. Delež variance za skupno okolje v gnezdu je v

fenotipski varianci predstavljal med 1 in 6 %. Majhen delež pojasnjene variabilnosti s tem vplivom

so lahko posledica relativno pogostega prestavljanja pujskov. Prestavljanje pujskov ob rojstvu zaradi


izenacevanja gnezd je obicajen postopek na farmah v Sloveniji, vendar pa se podatkov oprestavljenju

pujskov in njihovem številu ne beleži.

Ocene deleža variabilnosti za skupno okolje v gnezdu so manjše v srednjih zaporednih prasitvah,

obicajno v tretji in cetrti zaporedni prasitvi. Najvecji delež variance je imel ta vpliv na vseh treh

farmah v šesti zaporedni prasitvi. V prvih zaporednih prasitvah je bilo v podatkih vkljucenih vec

svinj iz istega gnezda kot kasneje, poleg tega pa je ta komponenta dokaj majhna in se lahko njen delež

v primerjavi s fenotipsko varianco precej spreminja. Pri veclastnostni analizi vpliva permanentnega

okolja znotraj svinje ni bilo mogoce lociti od ostanka, saj vsako gnezdo predstavlja svojo lastnost z

le eno meritvijo na svinjo. Ocene za variance ostanka kot deleža fenotipske variance po zaporednih

prasitvah so bile podobne med farmami in so zavzemale vrednosti med 0.82 in 0.88.

Ocene korelacij za skupno okolje v gnezdu so bile manjše v primerjavi s korelacijami za aditivni

genetski vpliv in so se precej spreminjale med pari zaporednih prasitev. Podobne vzorce pri nihanju

korelacij za skupno okolje v gnezdu smo spremljali na vseh farmah. Korelacije pri ostanku so bile

nedvomno najmanjše in so bile vecinoma manjše od 0.10. Te majhne korelacije dokazujejo, da smo

z vkljucitvijo nakljucnih vplivov uspešno pojasnili vecino povezave med gnezdi po zaporednih pra-

sitvah. Veclastnostno analizo smo opravili le za manjši niz podatkov DS1, saj so se zaradi visokih

genetskih in okoljskih korelacij med sosednjimi višjimi zaporednimi prasitvami, pojavile numericne

težave pri izracunih za niz podatkov DS2, ki znižujejo rang matrik komponent kovariance.

Model z nakljucno regresijo je bil uporabljen tako za niz podatkov do šesteprasitve kot za niz s

prasitvami do desete zaporedne prasitve. Genetskim in okoljskim kovariancnim funkcijam, kjer smo

uporabili Legendrove polinome od prve do tretje stopnje, smo izracunali lastne vrednosti. Le-te

prikažejo relativni pomen za vsak dodaniclen Legendrovih polinomov.

Vsota lastnih vrednosti za direktno aditivno genetsko in permanentno okolijsko varianco se je po-

vecevala s stopnjo Legendrovih polinomov, medtem ko je vsota lastnih vrednosti za skupno okolje

v gnezdu ostajala podobna, ne glede na model. Na splošno se jeprva lastna vrednost, ki vecinoma

pripada konstantnemuclenu polinoma, povecevala za vse nakljucne vplive v obeh nizih podatkov

s povecevanjem stopnje Legendrovih polinomov, njen delež v skupni variabilnosti pa se je zmanj-

ševal. Prve tri lastne vrednosti so pojasnile vec kot 99 % skupne variabilnosti za direktni aditivni

genetski vpliv in vpliv skupnega okolja v gnezdu. V primeru vpliva permanentnega okolja sta bili

dovolj prvi dve lastni vrednosti za pojasnitev prakticno celotne variabilnosti. Kljub temu, da smo

uporabili Legendrove polinome enake stopnje za vse nakljucne vplive, z namenom, da bi zagotovili

enako možnost izražanja variabilnosti pri vseh, se je pokazalo, da bi bili za nekatere nakljucne vplive

primernejši polinomi nižjih stopenj.

Kvadratni Legendrov polinom, ki ima tri regresijske koeficiente, je zadošcal pri modeliranju varia-

bilnosti velikosti gnezda po zaporednih prasitvah do šesteprasitve v nizu DS1, saj zajame prakticno

vso variabilnost. Lastne vrednosti genetskih kovariancnih funkcij za niz podatkov DS1 dokazujejo,

da variabilnost konstantnegaclena Legendrovega polinoma pojasnjuje med 90 in 95 % genetske vari-

abilnosti. Vkljucitev višjih zaporednih prasitev v podatke (DS2) je imela zaposledico, da se je delež

genetske variance, ki jo pokriva konstantniclen, zmanjšal na 85 do 90 %. Preostalih 10 do 15 %


genetske variabilnosti je zajetih v individualnih proizvodnih krivuljah svinj. Trend lahko pojasnimo

na ta nacin, da polinomi bolje opišejo spreminjanje velikosti gnezda po zaporednih prasitvah kot zgolj

nivo oz. povprecje v ponovljivostnem modelu živali. Spremembe so bolj izrazite po šesti zaporedni

prasitvi.

Delež direktne aditivne genetske variance za Legendrove polinome višjih stopenj je bil vecinoma do-

bro opisan z linarno in kvadratno regresijo. V podatkih nizaDS1 so se po farmah linearni koeficienti

gibali med 4.18 % in 10.03 %, medtem ko so linearni koeficientiniza DS2 po farmah zavzemali med

7.65 % in 14.20 % variabilnosti. Kvadratni regresijski koeficienti, ki so opisali direktno aditivno

genetsko varianco, so zavzeli vrednosti od 0.52 % do 1.47 % v nizu DS1. V nizu DS2 je bil delež

variance za direktni aditivni genetski vpliv, ki ga zajema kvadratni regresijski koeficient, vecji v pri-

merjavi z nizom DS1, in se je gibal med 1.50 in 3.75 %. Delež pojasnjene genetske variabilnosti,

ki jo je prispeval kubni regresijski koeficient, je bil v skupni variabilnosti vseh nakljucnih vplivov

zanemarljivo majhen.

Delež variance za vpliv permanentnega okolja, ki ga je pojasnjeval kontantniclen polinoma, se je

podobno zmanjšal z dodatkom meritev do desete zaporedne prasitve v niz podatkov. Zmanjšanje je

bilo ocitnejše kot pri direktnem aditivnem genetskem vplivu. Konstantniclen za vpliv permanentnega

okolja je pojasnil med 70 in 95 % variabilnosti, kar pomeni, da je bilo do 30 % celotne variabilnosti

permanentnega okolja pojasnjenega z odstopanjem individualnih krivulj svinj. Opažena variabilnost

konstantnegaclena za skupno okolje v gnezdu se je s povecevanjem stopnje Legendrovih polino-

mov spreminjala in razlikovala med farmami. Zaradi majhnihdeleža vpliva permanentnega okolja v

fenotipski varianci, te spremembe niso tako pomembne.

Ocenili smo genetske lastne funkcije za model z nakljucno regresijo s kubnimi Legendrovimi poli-

nomi. Prva lastna funkcija, ki je pojasnila med 85 % in 90 % genetske variance za število živorojenih

pujskov, je bila na celotnemcasovnem intervalu pozitivna in bolj ali manj konstantna odprve do de-

sete zaporedne prasitve. To pomeni, da bi selekcija na velikost gnezda v katerikoli zaporedni prasitvi

povzrocila podoben odziv za vse prasitve. Velikost prve lastne vrednosti kaže, da je lahko selekcija

na prvo komponento uspešna. Druga lastna funkcija, ki pojasni med 7 % in 14 % genetske variance,

zamenja svoj predznak po peti zaporedni prasitvi, kar pomeni, da bi selekcijski pritisk na povecanje

velikosti gnezda do pete prasitve imel za posledico zmanjševanje velikosti gnezda po peti prasitvi

in obratno, selekcija na povecano velikost gnezda po peti prasitvi bi povzrocila zmanjšanje velikosti

gnezda v nižjih prasitvah. S tem bi dosegli genetske spremembe v obliki krivulje. Kljub temu pa veli-

kost druge lastne vrednosti pokaže, da je bi bil odziv na selekcijo, ki vkljucuje drugo lastno funkcijo,

pocasnejši v primerjavi s spremembami, ki vkljucujejo prvo komponento. Delež tretje incetrte lastne

vrednosti je bil zanemarljivo majhen, zato potek krivulj lastnih funkcij in njihova vloga pri selekciji

na velikost gnezda nista pomembni.

Z modelom z nakljucno regresijo ocenjene genetske in okoljske variance, kot tudi njihovi deleži v

skupni fenotipski varianci, so bili za niz podatkov DS1 podobni rezultatom iz veclastnostne analize.

Ocene deleža v fenotipski varianci za manjši in vecji niz podatkov so bile podobne za prvih šest

prasitev. Kljub temu, da se pogosto omenjajo omejitve modelov z nakljucno regresijo za ocene na


robovih casovnega intervala, analiza z nakljucno regresijo na vecjem nizu podatkov potrjuje ocene,

dobljene na podatkih do vkljucno šeste zaporedne prasitve. Po osmi zaporedni prasitvi seocene

deležev nakljucnih komponent v fenotipski varianci bistveno spreminjajo. Pri presoji rezultatov za

višje zaporedne prasitve v nizu podatkov DS2 je potrebno nekoliko previdnosti, saj je gnezd iz devete

ali desete zaporedne prasitve bolj malo. To majhno število podatkov pomeni malo informacij pri

ocenjevanju robov kovariancne funkcije, zato so le-ti ocenjeni manj zanesljivo. Polegtega pa v višjih

prasitvah ostajajo v reprodukciji le svinje z boljšimi rezultati. Nenavadno obnašanje kovarianc na

robovih opazovanegacasovnega intervala so opazili tudi pri drugih lastnostih.Nihce pa se še ni lotil

raziskave, ali bolje podatke iz obdelave izkljuciti ali pa zadržati in pri selekciji uporabiti le zanesljivo

ocenjen krajši interval.

Komponente kovariance za število živorojenih pujskov, ki smo jih ocenili s pomocjo modela z na-

kljucno regresijo, so se podobno kot pri veclastnostnem modelu spreminjale z zaporedno prasitvijo.

Delež variabilnosti, ki jo je pojasnila oblika individualnih genetskih proizvodnih krivulj, se je z vklju-

cevanjem višjih prasitev povecevala. Pokritje 10 do 15 % genetske variabilnosti z obliko krivulje

kaže, da uporaba modelov z nakljucno regresijo predstavlja podlago za selekcijo tako na nivokot na

obliko krivulje za velikost gnezda po zaporedni prasitvah.

Ocenjene genetske korelacije med velikostjo gnezda po zaporednih prastvah z uporabo modela z

nakljucno regresijo kažejo podoben trend kot ocene, dobljene s veclastnostno analizo. Genetske ko-

relacije med številom živorojenih pujskov v razlicnih prasitvah so bile najvecje med sosednimi zapo-

rednimi prasitvami. Aditivne genetske korelacije so se zmanjševale z oddaljenostjo med zaporednimi

prasitvami. Ocene genetskih korelacij, pridobljene z modelom z nakljucno regresijo so bile malenkost

vecje za vecino izmed prvih šestih prasitev v primerjavi z ocenami iz veclastnostne analize. Na splo-

šno so se direktne aditivne genetske korelacije med prvo in višjimi zaporednimi prasitvami gibale od

0.40 do 0.90. Struktura genetskih korelacij kaže, da velikost gnezda v razlicnih prasitvah ni genetsko

povsem ista lastnost, saj so genetske korelacije za velikost gnezda v medseboj oddaljenih prasitvah

bistveno manjše od ena. Genetske korelacije dokazujejo obstoj individualne genetske variabilnosti,

kot tudi opravicljivost uporabe pristopa z nakljucno regresijo v genetskem vrednotenju.

Ocene korelacij za vpliv skupnega okolja v gnezdu so bile medsosednjimi zaporednimi prasitvami

zelo visoke, preko 0.90. Na splošno so se korelacije na vseh treh farmah zmanjševale s povecevanjem

oddaljenosti med zaporednimi prasitvami, vendar pa je bil ta vzorec zmanjševanja obcasno prekinjen

in so se korelacije kasneje spet povecale.

Ocene fenotipskih korelacij za velikost gnezda med razlicnimi zaporednimi prasitvami so bile med

0.06 in 0.37. Na vseh farmah so bili rezultati zelo podobni. Korelacije med prvo in višjimi prasitvami

so zavzemale vrednosti od 0.18 med prvo in drugo zaporedno prasitvijo do 0.06 med prvo in deseto

prasitvijo. Med sosednimi zaporednimi prasitvami so bile fenotipske korelacije pricakovano višje,

povecavale so se do 0.26 na farmi C oziroma do 0.37 na farmi A. Ocenena osnovi modela z nakljucno

regresijo so kazale podoben vzorec kot vrednosti, ki smo jihocenili z veclastnostno analizo za prvih

šestih zaporednih prasitev.

Rešitve za nakljucne regresijske koeficiente za živali, pricemer smo vkljucili Legendrove polinome


tretje stopnje, smo uporabili za napoved plemenskih vrednosti za število živorojenih pujskov po zapo-

rednih prasitvah oziroma za napoved genetskih proizvodnihfunkcij za posamezne živali. Med njimi

obstajajo precejšne razlike. Genetske proizvodne funkcije smo na osnovi razlik uvrstili v dve skupini,

živalmi so se razlikovale v nivoju kot tudi v obliki krivulj.Krivulje smo glede na specificen potek

razdelili v dve skupini. Prva skupina vkljucuje krivulje, ki narašcajo docetrte oziroma do pete zapo-

redne prasitve in potem padajo v višjih prasitvah. Krivuljev drugi skupini najprej padajo in kasneje

narašcajo. Znotraj obeh teh skupin pa se genetske proizvodne krivulje razlikujejo v tocki prevoja

oziroma v zaporedni prasitvi, kjer krivulja spremeni smer.Ni povsem enostavno predstavljivo, da se

genetska vrednost živali za velikost gnezda spreminja z zaporedno prasitvijo. Na splošno je znano,

da velikost gnezda narašca docetrte oziroma pete prasitve in kasneje pada. Razlicna genetska vre-

dnost v razlicnih zaporednih prasitvah pa ne pomeni nic drugega, kot da žival v razlicnih zaporednih

prasitvah razlicno odstopa od povprecja populacije. Obstajajo fiziološki mehanizmi, ki se vkljucujejo

in izkljucujejo med staranjem svinje in so po vsej verjetnosti tudi genetsko kontrolirani. Postavlja se

vprašanje, katera oblika krivulje bi bila za selekcijo najbolj zaželena. Izbira živali, kateri napoved

plemenske vrednosti za velikost gnezda bi narašcala z zaporednimi prasitvami, bi verjetno imela za

posledico izboljšanja perzistence. Za selekcijo so zanimive živali z višjim nivojem oz. povprecjem

napovedi preko celotnegacasovnega intervala, ki se pokaže že pri prvih dveh prasitvah in vztraja tudi

pri višjih.

V primerjavi z veclastnostnim modelom, je model z nakljucno regresijo s statisticne plati pravilnejši,

saj omogoca vkljucitev permanentnega okolja svinje, ki ima znaten prispevekv deležu pojasnjene

fenotipske variabilnosti. Tudi ponovljivostni model vkljucuje permanentno okolje, vendar pa ne omo-

goca spreminjanja korelacij med zaporednimi prasitvami.Ce primerjamo vse tri modele med sabo

opazimo trend narašcanja deleža pojasnjene variabilnosti od ponovljivostnega modela preko vecla-

stnostnega do modela z nakljucno regresijo, veca pa se tudi heritabiliteta.

Po numericni oz. racunalniški plati so modeli z nakljucno regresijo stabilnejši (robustnejši) v pri-

merjavi z veclastnostnimi, saj pri njih numericno nestabilnost povzrocajo korelacije med lastnostmi

z vrednostjo blizu ena, ki znižujejo rang matrik kovariance. Za ocenjevanje genetskih parametrov za

velikost gnezda model z nakljucno regresijo porabi manj racunalniškegacasa v primerjavi z vecla-

stnostnim modelom. Dodatno se je model z nakljucno regresijo izkazal za robustnejšega, saj smo

pri veclastnostnem modelu za dosego globalnega maksimuma logaritma verjetnostne funkcije po-

trebovali vec ponovitev iz razlicnih startnih vrednosti pri komponentah kovariance, nismopa uspeli

vkljuciti višjih zaporednih prasitev.

Nenazadnje so modeli z nakljucno regresijo tudi elegntnejši pri dolocanju kriterijev selekcije. Napo-

vedi plemenske vrednosti po zaporednih prasitvah bi lahko združili v skupno napoved z ekonomskimi

težami. Poleg tega lahko uporabimo za selekcijo izpeljane napovedi, ki opisujejo posebnosti proizvo-

dnih funkcij, npr. nivo, vztrajnost oz. perzistenco ter obliko krivulje. Pred uporabo v praksi pa bo

potrebno postopek dodelati še z aplikativnega vidika.


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ACKNOWLEDGEMENTS

I would like to express my gratitude to my supervisor Prof. Dr. Milena Kovac for her guidance

and help with this dissertation as well as for the encouragement during my study. I would also like

to thank my other two committee members Prof. Dr. Marija Uremovic and Prof. Dr. Jurij Pohar

for their patience, understanding and useful comments to this work. Thanks to Dr. Špela Malovrh

for her help with Linux, programming and other technical problems. Thanks to the Slovenian pig

breeding organization and farms for providing data which enabled my work. I would like to thank

Mrs. Karmela Malinger for her effort with reading the English manuscript. Thanks to all my co-

workers in Domžale for their support, comradeship, and goodwishes during my work, especially to

Dragutin Vincek, Gregor Gorjanc, Vesna Gantner, Dragan Radojkovic, Tina Flisar, Marija Špehar

and Marjeta Furman. I would also like to thank my co-workers in Zagreb for taking over my non-

research and research work. Finally, I wish to express my deepest gratitude for the constant support,

understanding and love that I received from my wife Sanja andmy children, Leon and Lukas, during

the last years.

ZAHVALA

Zahvaljujem se mentorici prof. dr. Mileni Kovac za nasvete in pomoc pri doktorskem delu ter za

vzpodbude med delom. Zahvaljujem se tudi ostalima dvemaclanoma komisije za oceno in zagovor

prof. dr. Mariji Uremovic in prof. dr. Juriju Poharju za potrpežljivost, razumevanje in koristne

pripombe. Hvala asist. dr. Špeli Malovrh za pomoc pri Linuxu, programiranju in ostalih tehnicnih

zadevah. Hvala slovenski prašicerejski organizaciji in farmam, ki so mi odstopili podatkein s tem

omogocili izvedbo naloge. Zahvaljujem se ga. Karmeli Malinger zaves njen trud pri lektoriranju an-

gleškega jezika v nalogi. Hvala vsem sodelavkam in sodelavcem na Katedri za etologijo, biometrijo in

selekcijo ter prašic erejo za podporo, družbo in dobre želje vcasu študija, posebno Dragutinu Vinceku,

Gregorju Gorjancu, Vesni Gantner, Draganu Radojkovicu, Tini Flisar, Mariji Špehar in Marjeti Fur-

man. Zahvaljujem se tudi vsem mojim sodelavkam in sodelavcem v Zavodu za specijalno stocarstvo

na Agronomski fakulteti v Zagrebu, ker so mi v zadnjem obdobju pomagali pri neraziskovalnih in

raziskovalnih obveznostih. Na koncu, želim izraziti zahvalo soprogi Sanji in sinovoma, Leonu in

Lukasu, za stalno podporo, razumevanje in ljubezen, katerosam sprejemal v zadnjih letih.

covariance functions for litter size in …member: prof. dr. marija uremovic´ university of zagreb,...

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